A097889 -id:A097889

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A266955

Intersection of A046346 (numbers that are divisible by the sum of their prime factors, counted with multiplicity) and A097889 (numbers that are products of at least two consecutive primes).

+20
1

30, 105, 15015, 9699690, 37182145, 215656441, 955049953, 33426748355, 247357937827, 1448810778701, 3710369067405, 304250263527210, 102481630431415235, 1086305282573001491, 261682369333342226303, 37420578814667938361329, 241532826894674874877669

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Alladi and Erdős ask if this sequence is infinite and give 3 terms: 2*3*5, 2*3*5*7*11*13*17*19 and 2*3*5*7*11*13*17*19*23*29*31*37*41, that is, a(1), a(4) and a(12).

This sequence contains A159578(n) for all values of n > 1. - Altug Alkan, Jan 07 2016

LINKS

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500

K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.

PROG

(PARI) sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); }

list(lim)= {my(v=List(), p, t); for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1, e-1, prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim, next(2)); if (! (t % sopfr(t)), listput(v, t)); p=nextprime(p+1))); vecsort(Vec(v)); } \\ adapted from A097889

CROSSREFS

Cf. A046346, A097889, A159578.

KEYWORD

nonn

AUTHOR

Michel Marcus, Jan 07 2016

EXTENSIONS

a(13)-a(17) from Hiroaki Yamanouchi, Jan 12 2016

STATUS

approved

A006094

Products of 2 successive primes.
(Formerly M4110)

+10
142

6, 15, 35, 77, 143, 221, 323, 437, 667, 899, 1147, 1517, 1763, 2021, 2491, 3127, 3599, 4087, 4757, 5183, 5767, 6557, 7387, 8633, 9797, 10403, 11021, 11663, 12317, 14351, 16637, 17947, 19043, 20711, 22499, 23707, 25591, 27221, 28891, 30967, 32399, 34571, 36863

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The Huntley reference would suggest prefixing the sequence with an initial 4 - Enoch Haga. [But that would conflict with the definition! - N. J. A. Sloane, Oct 13 2009]

Sequence appears to coincide with the sequence of numbers n such that the largest prime < sqrt(n) and the smallest prime > sqrt(n) divide n. - Benoit Cloitre, Apr 04 2002

This is true: p(n) < [ sqrt(a(n)) = sqrt(p(n)*p(n+1)) ] < p(n+1) by definition. - Jon Perry, Oct 02 2013

a(n+1) = smallest number such that gcd(a(n), a(n+1)) = prime(n+1). - Alexandre Wajnberg and Ray Chandler, Oct 14 2005

Also the area of rectangles whose side lengths are consecutive primes. E.g., the consecutive primes 7,11 produce a 7 X 11 unit rectangle which has area 77 square units. - Cino Hilliard, Jul 28 2006

a(n) = A001358(A172348(n)); A046301(n) = lcm(a(n), a(n+1)); A065091(n) = gcd(a(n), a(n+1)); A066116(n+2) = a(n+1)*a(n); A109805(n) = a(n+1) - a(n). - Reinhard Zumkeller, Mar 13 2011

See A209329 for the sum of the reciprocals. - M. F. Hasler, Jan 22 2013

A078898(a(n)) = 3. - Reinhard Zumkeller, Apr 06 2015

REFERENCES

H. E. Huntley, The Divine Proportion, A Study in Mathematical Beauty. New York: Dover, 1970. See Chapter 13, Spira Mirabilis, especially Fig. 13-5, page 173.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

A. Bernoff and R. Pennington, Problems Drive 1984, Archimedeans Problems Drive, Eureka, 45 (1985), 22-25, 50. (Annotated scanned copy)

C. Cobeli and A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11 (2014) pp. 1489-1501, DOI: 10.1080/10236198.2014.940337. Also available as arXiv:1411.1334 [math.NT], 2014.

FORMULA

A209329 = Sum_{n>=2} 1/a(n). - M. F. Hasler, Jan 22 2013

a(n) = A000040(n) * A000040(n+1). - Alois P. Heinz, Jan 02 2021

MAPLE

a:= n-> (p-> p(n)*p(n+1))(ithprime):

seq(a(n), n=1..43); # Alois P. Heinz, Jan 02 2021

MATHEMATICA

Table[ Prime[n] Prime[n + 1], {n, 40}] (* Robert G. Wilson v, Jan 22 2004 *)

Times@@@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Oct 15 2011 *)

PROG

(PARI) g(n) = for(x=1, n, print1(prime(x)*prime(x+1)", ")) \\ Cino Hilliard, Jul 28 2006

(PARI) is(n)=my(p=precprime(sqrtint(n))); p>1 && n%p==0 && isprime(n/p) && nextprime(p+1)==n/p \\ Charles R Greathouse IV, Jun 04 2014

(MuPAD) ithprime(i)*ithprime(i+1) $ i = 1..41 // Zerinvary Lajos, Feb 26 2007

(Magma) [NthPrime(n)*NthPrime(n+1): n in [1..41]]; // Bruno Berselli, Feb 24 2011

(Haskell)

a006094 n = a006094_list !! (n-1)

a006094_list = zipWith (*) a000040_list a065091_list

-- Reinhard Zumkeller, Mar 13 2011

(Haskell)

a006094_list = pr a000040_list

where pr (n:m:tail) = n*m : pr (m:tail)

pr _ = []

-- Jean-François Antoniotti, Jan 08 2020

(Python)

from sympy import prime, primerange

def aupton(nn):

alst, prevp = [], 2

for p in primerange(3, prime(nn+1)+1): alst.append(prevp*p); prevp = p

return alst

print(aupton(43)) # Michael S. Branicky, Jun 15 2021

CROSSREFS

Subset of the squarefree semiprimes, A006881.

Cf. A090076, A090090.

Cf. A166329, A152241, A030664, A219603.

Cf. A046301, A046302, A046303, A046324, A046325, A046326, A046327.

Subsequence of A256617 and A097889.

Cf. A000040, A078898.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A073485

Product of any number of consecutive primes; squarefree numbers with no gaps in their prime factorization.

+10
35

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 221, 223, 227, 229, 233

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A073484(a(n)) = 0 and A073483(a(n)) = 1;

See A097889 for composite terms. - Reinhard Zumkeller, Mar 30 2010

A169829 is a subsequence. - Reinhard Zumkeller, May 31 2010

a(A192280(n)) = 1: complement of A193166.

Also fixed points of A053590: a(n) = A053590(a(n)). - Reinhard Zumkeller, May 28 2012

The Heinz numbers of the partitions into distinct consecutive integers. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} prime(p_j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: (i) 15 (= 3*5) is in the sequence because it is the Heinz number of the partition [2,3]; (ii) 10 (= 2*5) is not in the sequence because it is the Heinz number of the partition [1,3]. - Emeric Deutsch, Oct 02 2015

Except for the term 1, each term can uniquely represented as A002110(k)/A002110(m) for k > m >= 0; 1 = A002110(k)/A002110(k) for all k. - Michel Marcus and Jianing Song, Jun 19 2019

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

FORMULA

a(n) ~ n log n. - Charles R Greathouse IV, Oct 24 2012

EXAMPLE

105 is a term, as 105 = 3*5*7 with consecutive prime factors.

MAPLE

isA073485 := proc(n)

local plist, p, i ;

plist := sort(convert(numtheory[factorset](n), list)) ;

for i from 1 to nops(plist) do

p := op(i, plist) ;

if modp(n, p^2) = 0 then

return false;

end if;

if i > 1 then

if nextprime(op(i-1, plist)) <> p then

return false;

end if;

end do:

true;

end proc:

for n from 1 to 1000 do

if isA073485(n) then

printf("%d, ", n);

end if;

end do: # R. J. Mathar, Jan 12 2016

# second Maple program:

q:= proc(n) uses numtheory; n=1 or issqrfree(n) and (s->

nops(s)=1+pi(max(s))-pi(min(s)))(factorset(n))

end:

select(q, [$1..288])[]; # Alois P. Heinz, Jan 27 2022

MATHEMATICA

f[n_] := FoldList[ Times, 1, Prime[ Range[n, n + 3]]]; lst = {}; k = 1; While[k < 55, AppendTo[lst, f@k]; k++ ]; Take[ Union@ Flatten@ lst, 65] (* Robert G. Wilson v, Jun 11 2010 *)

PROG

(Haskell)

a073485 n = a073485_list !! (n-1)

a073485_list = filter ((== 1) . a192280) [1..]

-- Reinhard Zumkeller, May 28 2012, Aug 26 2011

(PARI) list(lim)=my(v=List(primes(primepi(lim))), p, t); for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1, e-1, prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim, next(2)); listput(v, t); p=nextprime(p+1))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 24 2012

CROSSREFS

Complement: A193166.

Intersection of A005117 and A073491.

Subsequence of A277417.

Cf. A053590, A073483, A073484, A073486, A192280.

Cf. A000040, A006094, A002110, A097889, A169829 (subsequences).

Cf. A096334.

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Aug 03 2002

EXTENSIONS

Alternative description added to the name by Antti Karttunen, Oct 29 2016

STATUS

approved

A050936

Sum of two or more consecutive prime numbers.

+10
28

5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 36, 39, 41, 42, 48, 49, 52, 53, 56, 58, 59, 60, 67, 68, 71, 72, 75, 77, 78, 83, 84, 88, 90, 95, 97, 98, 100, 101, 102, 109, 112, 119, 120, 121, 124, 127, 128, 129, 131, 132, 138, 139, 143, 144, 150, 152, 155, 156, 158, 159, 160, 161, 162

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Patrick De Geest, WONplate 122

Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Eric Weisstein's World of Mathematics, Prime Sums

EXAMPLE

E.g., 5 = (2 + 3) or (#2,2).

2+3 = 5, 3+5 = 8, 2+3+5 = 10, 7+5 = 12, 3+5+7 = 15, etc.

MAPLE

# uses code of A084143

isA050936 := proc(n::integer)

if A084143(n) >= 1 then

true;

else

false;

end if;

end proc:

for n from 1 to 300 do

if isA050936(n) then

printf("%d, ", n);

end if;

end do: # R. J. Mathar, Aug 19 2020

MATHEMATICA

lst={}; Do[p=Prime[n]; Do[p=p+Prime[k]; AppendTo[lst, p], {k, n+1, 2*10^2}], {n, 2*10^2}]; Take[Union[lst], 10^2] (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

f[n_] := Block[{len = PrimePi@ n}, p = Prime@ Range@ len; Count[ Flatten[ Table[ p[[i ;; j]], {i, len}, {j, i+1, len}], 1], q_ /; Total@ q == n]]; Select[ Range@ 150, f@ # > 0 &] (* Or quicker for a larger range *)

lmt = 150; p = Prime@ Range@ PrimePi@ lmt; t = Table[0, {lmt}]; Do[s = 0; j = i+1; While[s = s + p[[j]]; s <= lmt, t[[s]]++; j++], {i, Length@ p}]; Select[ Range@ lmt, t[[#]] > 0 &] (* Robert G. Wilson v *)

Module[{nn=70, prs}, prs=Prime[Range[nn]]; Take[Union[Flatten[Table[Total/@ Partition[prs, i, 1], {i, 2, nn}]]], nn]] (* Harvey P. Dale, Nov 13 2013 *)

PROG

(Haskell)

import Data.Set (empty, findMin, deleteMin, insert)

import qualified Data.Set as Set (null)

a050936 n = a050936_list !! (n-1)

a050936_list = f empty [2] 2 $ tail a000040_list where

f s bs c (p:ps)

| Set.null s || head bs <= m = f (foldl (flip insert) s bs') bs' p ps

| otherwise = m : f (deleteMin s) bs c (p:ps)

where m = findMin s

bs' = map (+ p) (c : bs)

-- Reinhard Zumkeller, Aug 26 2011

(PARI) is(n)=my(v, m=1, t); while(1, v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1, m, v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1, 1, -1, v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(0)); t=vecsum(v); if(t==n, return(1)); if(t>n, while(t>n, t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(t<n, t-=v[1]; v=concat(v[2..m], nextprime(v[m]+1)); t+=v[m])); if(v[1]==0, return(0)); if(t==n, return(1))) \\ Charles R Greathouse IV, May 05 2016

(PARI) list(lim)=my(v=List(), s, n=1, p); while(1, p=2; s=vecsum(primes(n++)); if(s>lim, return(Set(v))); listput(v, s); forprime(q=prime(n+1), , s+=q-p; if(s>lim, break); listput(v, s); p=nextprime(p+1))); \\ Charles R Greathouse IV, Nov 24 2021

CROSSREFS

Subsequence of A034707.

Cf. A067372 up to A067381, A054996, A000040.

A084143(a(n)) > 0, complement of A087072.

Cf. A054845, A097889.

KEYWORD

nice,nonn,easy

AUTHOR

G. L. Honaker, Jr., Dec 31 1999

EXTENSIONS

More terms from David W. Wilson, Jan 13 2000

STATUS

approved

A066312

Numbers, other than prime powers, whose distinct prime factors are consecutive primes.

+10
20

6, 12, 15, 18, 24, 30, 35, 36, 45, 48, 54, 60, 72, 75, 77, 90, 96, 105, 108, 120, 135, 143, 144, 150, 162, 175, 180, 192, 210, 216, 221, 225, 240, 245, 270, 288, 300, 315, 323, 324, 360, 375, 384, 385, 405, 420, 432, 437, 450, 480, 486

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers whose squarefree kernel (A007947) is the product of 2 or more consecutive primes. - Peter Munn, May 27 2023

LINKS

Harry J. Smith, Table of n, a(n) for n=1..1000

FORMULA

{a(n) : n >= 1} = {k >= 1 : A007947(k) is in A097889}. - Peter Munn, May 29 2023

EXAMPLE

75 is included because 75 = 3 * 5^2 and 3 and 5 are consecutive primes.

MATHEMATICA

Select[Range[2, 500], And[! PrimePowerQ@ #, Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] (* Michael De Vlieger, Sep 24 2017 *)

PROG

(PARI) { n=0; for (m=2, 10^9, f=factor(m); b=1; if (matsize(f)[1] == 1, next); for (i=2, matsize(f)[1], if (primepi(f[i, 1]) - primepi(f[i - 1, 1]) > 1, b=0; break)); if (b, write("b066312.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 10 2010

CROSSREFS

Cf. A007947, A097889.

KEYWORD

nonn

AUTHOR

Leroy Quet, Jan 01 2002

EXTENSIONS

OFFSET changed from 0,1 to 1,1 by Harry J. Smith, Feb 10 2010

Name edited by Peter Munn, May 29 2023

STATUS

approved

A045619

Numbers that are the products of 2 or more consecutive integers.

+10
17

0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - T. D. Noe, Oct 13 2002

Numbers of the form x!/y! with y+1 < x. - Reinhard Zumkeller, Feb 20 2008

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T.D. Noe)

P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292-301.

FORMULA

a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - Reinhard Zumkeller, Feb 27 2008

Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012

a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - Charles R Greathouse IV, Aug 27 2013

EXAMPLE

30 is in the sequence as 30 = 5*6 = 5*(5+1). - David A. Corneth, Oct 19 2021

MATHEMATICA

maxNum = 1700; lst = {}; For[i = 1, i <= Sqrt[maxNum], i++, j = i + 1; prod = i*j; While[prod < maxNum, AppendTo[lst, prod]; j++; prod *= j]]; lst = Union[lst]

PROG

(Python)

import heapq

from sympy import sieve

def aupton(terms, verbose=False):

p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}

while len(aset) < terms:

(v, s, l) = heapq.heappop(h)

aset.add(v)

if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i]")

if v >= p:

p *= nextcount

heapq.heappush(h, (p, 2, nextcount))

nextcount += 1

v //= s; s += 1; l += 1; v *= l

heapq.heappush(h, (v, s, l))

return sorted(aset)

print(aupton(52)) # Michael S. Branicky, Oct 19 2021

(PARI) list(lim)=my(v=List([0]), P, k=1, t); while(1, k++; P=binomial('n+k-1, k)*k!; if(subst(P, 'n, 1)>lim, break); for(n=1, lim, t=eval(P); if(t>lim, next(2)); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Nov 16 2021

CROSSREFS

Union of A002378, A007531, A052762, A052787, A053625, etc. - R. J. Mathar, Oct 19 2021

Cf. A001597, A000142, A137895, A093449, A064224, A084720, A137899, A137900, A120436, A097889.

KEYWORD

easy,nonn,nice

AUTHOR

Erich Friedman

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000

More terms from Reinhard Zumkeller, Feb 27 2008

Incorrect program removed by David A. Corneth, Oct 19 2021

STATUS

approved

A073490

Number of prime gaps in factorization of n.

+10
17

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,110

COMMENTS

A137723(n) is the smallest number of the first occurring set of exactly n consecutive numbers with at least one prime gap in their factorization: a(A137723(n)+k)>0 for 0<=k<n and a(A137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..50000

FORMULA

a(n) = A073484(A007947(n)).

a(A000040(n))=0; a(A000961(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.

a(A073486(n))>0; a(A073487(n)) = 1; a(A073488(n))=2; a(A073489(n))=3.

a(n)=0 iff A073483(n) = 1.

a(A097889(n)) = 0. - Reinhard Zumkeller, Nov 20 2004

0 <= a(m*n) <= a(m) + a(n) + 1. A137794(n) = 0^a(n). - Reinhard Zumkeller, Feb 11 2008

EXAMPLE

84 = 2*2*3*7 with one gap between 3 and 7, therefore a(84) = 1;

110 = 2*5*11 with two gaps: between 2 and 5 and between 5 and 11, therefore a(110) = 2.

MAPLE

A073490 := proc(n)

local a, plist ;

plist := sort(convert(numtheory[factorset](n), list)) ;

a := 0 ;

for i from 2 to nops(plist) do

if op(i, plist) <> nextprime(op(i-1, plist)) then

a := a+1 ;

end if;

end do:

end proc:

seq(A073490(n), n=1..110) ; # R. J. Mathar, Oct 27 2019

MATHEMATICA

gaps[n_Integer/; n>0]:=If[n===1, 0, Complement[Prime[PrimePi[Rest[ # ]]-1], # ]&[First/@FactorInteger[n]]]; Table[Length[gaps[n]], {n, 1, 105}] (Wouter Meeussen, Oct 30 2004)

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Table[Total[pa @@@ Partition[First /@ FactorInteger[n], 2, 1]], {n, 120}] (* Jayanta Basu, Jul 01 2013 *)

PROG

(Haskell)

a073490 1 = 0

a073490 n = length $ filter (> 1) $ zipWith (-) (tail ips) ips

where ips = map a049084 $ a027748_row n

-- Reinhard Zumkeller, Jul 04 2012

(Python)

from sympy import primefactors, nextprime

def a(n):

pf = primefactors(n)

return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:]))

print([a(n) for n in range(1, 121)]) # Michael S. Branicky, Oct 14 2021

CROSSREFS

Cf. A073491, A073492, A073493, A073494, A073495.

Cf. A137721, A137722, A027748, A049084.

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Aug 03 2002

EXTENSIONS

More terms from Franklin T. Adams-Watters, May 19 2006

STATUS

approved

A257836

Numbers which are the product of at least two consecutive odd numbers > 1.

+10
2

15, 35, 63, 99, 105, 143, 195, 255, 315, 323, 399, 483, 575, 675, 693, 783, 899, 945, 1023, 1155, 1287, 1295, 1443, 1599, 1763, 1935, 2115, 2145, 2303, 2499, 2703, 2915, 3135, 3315, 3363, 3465, 3599, 3843, 4095, 4355, 4623, 4845, 4899, 5183, 5475, 5775, 6083

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

EXAMPLE

. | | ----- Factorizations into ... --------------

. n | a(n) | prime powers | consecutive odd numbers

. ----+-------+--------------------+--------------------------

. 1 | 15 | 3 * 5 | 3 * 5

. 2 | 35 | 5 * 7 | 5 * 7

. 3 | 63 | 3^2 * 7 | 7 * 9

. 4 | 99 | 3^2 * 11 | 9 * 11

. 5 | 105 | 3 * 5 * 7 | 3 * 5 * 7

. 6 | 143 | 11 * 13 | 11 * 13

. 7 | 195 | 3 * 5 * 13 | 13 * 15

. 8 | 255 | 3 * 5 * 17 | 15 * 17

. 9 | 315 | 3^2 * 5 * 7 | 5 * 7 * 9

. 10 | 323 | 17 * 19 | 17 * 19

. 11 | 399 | 3 * 7 * 19 | 19 * 21

. 12 | 483 | 3 * 7 * 23 | 21 * 23

. 13 | 575 | 5^2 * 23 | 23 * 25

. 14 | 675 | 3^3 * 5^2 | 25 * 27

. 15 | 693 | 3^2 * 7 * 11 | 7 * 9 * 11

. 16 | 783 | 3^3 * 29 | 27 * 29

. 17 | 899 | 29 * 31 | 29 * 31

. 18 | 945 | 3^3 * 5 * 7 | 3 * 5 * 7 * 9

. 19 | 1023 | 3 * 11 * 31 | 31 * 33

. 20 | 1155 | 3 * 5 * 7 * 11 | 33 * 35

. 21 | 1287 | 3^2 * 11 * 13 | 9 * 11 * 13

. 22 | 1295 | 5 * 7 * 37 | 35 * 37

. 23 | 1443 | 3 * 13 * 37 | 37 * 39

. 24 | 1599 | 3 * 13 * 41 | 39 * 41

. 25 | 1763 | 41 * 43 | 41 * 43

. 26 | 1935 | 3^2 * 5 * 43 | 43 * 45

. 27 | 2115 | 3^2 * 5 * 47 | 45 * 47

. 28 | 2145 | 3 * 5 * 11 * 13 | 11 * 13 * 15

. 29 | 2303 | 7^2 * 47 | 47 * 49

. 30 | 2499 | 3 * 7^2 * 17 | 49 * 51 .

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a257836 n = a257836_list !! (n-1)

a257836_list = f $ singleton (15, 3, 5) where

f s = y : f (insert (w, u, v') $ insert (w `div` u, u + 2, v') s')

where w = y * v'; v' = v + 2

((y, u, v), s') = deleteFindMin s

CROSSREFS

Cf. A005408, A097889.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 12 2015

STATUS

approved

A257891

Numbers that are products of at least three consecutive primes.

+10
2

30, 105, 210, 385, 1001, 1155, 2310, 2431, 4199, 5005, 7429, 12673, 15015, 17017, 20677, 30030, 33263, 46189, 47027, 65231, 82861, 85085, 96577, 107113, 146969, 190747, 215441, 241133, 255255, 290177, 323323, 347261, 392863, 409457, 478661, 510510, 583573

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

EXAMPLE

a(5) = 1001 = 7 * 11 * 13;

a(6) = 1155 = 3 * 5 * 7 * 11;

a(7) = 2310 = 2 * 3 * 5 * 7 * 11;

a(8) = 2431 = 11 * 13 * 17.

MATHEMATICA

Select[Module[{nn=1000}, Flatten[Table[Times@@@Partition[Prime[Range[nn]], d, 1], {d, 3, 7}]]]//Union, #<10^7&] (* Harvey P. Dale, Aug 04 2024 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a257891 n = a257891_list !! (n-1)

a257891_list = f $ singleton (30, 2, 5) where

f s = y : f (insert (w, p, q') $ insert (w `div` p, a151800 p, q') s')

where w = y * q'; q' = a151800 q

((y, p, q), s') = deleteFindMin s

CROSSREFS

Cf. A151800, A097889, A000977, A046301 (subsequence).

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 12 2015

STATUS

approved

A334175

Numbers that can be written as a product of two or more consecutive primorial numbers.

+10
2

2, 12, 180, 360, 6300, 37800, 75600, 485100, 14553000, 69369300, 87318000, 174636000, 14567553000, 15330615300, 437026590000, 2622159540000, 4951788741900, 5244319080000, 35413721343000, 2163931680210300, 7436881482030000, 148702215919257000, 223106444460900000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..1318

Eric Weisstein's World of Mathematics, Primorial.

Index entries for sequences related to primorial numbers.

EXAMPLE

2 = prime(0)# * prime(1)#;

12 = prime(1)# * prime(2)#;

180 = prime(2)# * prime(3)#;

360 = prime(1)# * prime(2)# * prime(3)#;