A061015 -id:A061015

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A092062

Numbers k such that A061015(k) is prime.

+20
1

2, 10, 18, 36, 90, 759

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

OFFSET

1,1

COMMENTS

a(6) > 447 for a(6) the numerator has more than 2673 digits.

a(7) > 1850. - Michael S. Branicky, Jun 27 2022

LINKS

Table of n, a(n) for n=1..6.

FORMULA

Numbers k such that numerator of (Sum_{i=1..k} 1/prime(i)^2) is prime

EXAMPLE

1/2^2 = 1/4 but 1 is not prime, 1/2^2 + 1/3^2 = 13/36 and 13 is prime so a(1)=2.

PROG

(PARI) sm(n)= s=0; for(i=1, n, s=s+1/(prime(i)^2)); return(s);

for (i=1, 400, if(isprime(numerator(sm(i))), print1(i, ", ")))

(Python) # uses A061015gen() and imports from A061015

from sympy import isprime

def agen():

yield from (k for k, ak in enumerate(A061015gen(), 1) if isprime(ak))

print(list(islice(agen(), 5))) # Michael S. Branicky, Jun 27 2022

CROSSREFS

Cf. A061015.

KEYWORD

hard,nonn

AUTHOR

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 20 2004

EXTENSIONS

a(6) from Alexander Adamchuk, Sep 16 2010

STATUS

approved

A024451

a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

+10
61

0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853

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

OFFSET

0,3

COMMENTS

Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002

(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011

Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014

Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018

Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022

The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)

Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

FORMULA

Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.

a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002

(n+1)-st elementary symmetric function of the first n primes.

a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006

From Antti Karttunen, Jan 31 2024 and Feb 08 2024: (Start)

a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).

For n > 0, a(n) = A327860(A143293(n-1)).

For n > 0, a(n) = A348301(n) + A002110(n).

For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]

(End)

EXAMPLE

0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...

MAPLE

h:= n-> add(1/(ithprime(i)), i=1..n);

t1:=[seq(h(n), n=0..50)];

t1a:=map(numer, t1); # A024451

t1b:=map(denom, t1); # A002110 - N. J. A. Sloane, Apr 25 2014

MATHEMATICA

a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a, 18] (* Jean-François Alcover, Apr 11 2011 *)

f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]

a[n_] := SymmetricPolynomial[n - 1, t[n]]

Table[a[n], {n, 1, 16}] (* A024451 *)

(* Clark Kimberling, Dec 29 2011 *)

Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)

PROG

(Magma) [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ]; // Bruno Berselli, Apr 11 2011

(PARI) a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018

(Python)

from sympy import prime

from fractions import Fraction

def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator

print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021

(Python)

from math import prod

from sympy import prime

def A024451(n):

q = prod(plist:=tuple(prime(i) for i in range(1, n+1)))

return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022

CROSSREFS

Denominators are A002110.

Row sums of A077011 and A258566.

Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.

Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).

Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes).

Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.

KEYWORD

nonn,frac,easy,nice

AUTHOR

N. J. A. Sloane, Clark Kimberling

EXTENSIONS

a(0)=0 prepended by Alois P. Heinz, Jun 26 2015

STATUS

approved

A115964

Denominator of Sum_{i=1..n} 1/prime(i)^3.

+10
11

8, 216, 27000, 9261000, 12326391000, 27081081027000, 133049351085651000, 912585499096480209000, 11103427767506874702903000, 270801499821725167129101267000, 8067447481189014453943055845197000

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

OFFSET

1,1

COMMENTS

Numerators are in A115963.

Also the primorials cubed. - Reikku Kulon, Sep 18 2008

LINKS

Table of n, a(n) for n=1..11.

FORMULA

a(n) = denominator of Sum_{i=1..n} 1/A000040(i)^3.

a(n) = A002110(n)^3. - Reikku Kulon, Sep 18 2008

EXAMPLE

1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.

MATHEMATICA

a[n_]:=Product[Prime[i]^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

CROSSREFS

Cf. A115963 (numerators).

Cf. A024451 (numerator of sum_{i=1..n} 1/prime(i)), A002110 (primorial, also denominator of sum_{i=1..n} 1/prime(i)), A061015 (numerator of sum_{i=1..n} 1/prime(i)^2).

Cf. A000040, A075986/A075987, A106830/A034386.

Cf. A061742, A100778. - Reikku Kulon, Sep 18 2008

KEYWORD

easy,frac,nonn

AUTHOR

Jonathan Vos Post, Mar 14 2006

STATUS

approved

A075986

Numerator of 1+1/prime(1)^2+ ... + 1/prime(n)^2 where prime(k) is the k-th prime.

+10
6

1, 5, 49, 1261, 62689, 7629469, 1294716361, 375074829229, 135662633811769, 71859617272521901, 60483708554835755641, 58166700851687469003901, 79670437976161330893757369, 133981073592392620630139873389

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

OFFSET

0,2

COMMENTS

The sum is similar to that in A061015 with an additional 1. The sum in the definition has limit about 1.45224742. The case of reciprocal cubes is in A075987.

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+Prime[i]^2 if i=j and 1 otherwise. - Alexander Adamchuk, Jul 08 2006

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 94-98.

LINKS

Table of n, a(n) for n=0..13.

Steven R. Finch, Meissel-Mertens Constants [Broken link]

Steven R. Finch, Meissel-Mertens Constants [From the Wayback machine]

FORMULA

a(0)=1; a(n)=a(n-1)*prime(n)^2+(prime(1)*...*prime(n-1))^2.

EXAMPLE

a(2) = 49 so a(3) = 49*p(3)^2 + (2*3)^2 = 49*25 + 36 = 1261.

MATHEMATICA

Table[Det[DiagonalMatrix[Table[Prime[i]^2, {i, 1, n}]]+1], {n, 1, 15}] (* Alexander Adamchuk, Jul 08 2006 *)

Accumulate[Join[{1}, 1/Prime[Range[20]]^2]]//Numerator (* Harvey P. Dale, May 08 2023 *)

CROSSREFS

Cf. A061015, A075987, A024528.

KEYWORD

nonn,frac

AUTHOR

Zak Seidov, Sep 28 2002

EXTENSIONS

Edited by Dean Hickerson, Sep 30 2002

STATUS

approved

A115963

Numerator of Sum_{i=1..n} 1/prime(i)^3.

+10
5

1, 35, 4591, 1601713, 2141141003, 4716413174591, 23198819007792583, 159253748925534977797, 1938552948676080555065099, 47290471293028435532185602511, 1409101231790431848106470385672201

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

OFFSET

1,2

COMMENTS

Denominators = A115964. See also: A024451 Numerator of Sum_{i=1..n} 1/prime(i). A002110 Primorial [denominator of Numerator of Sum_{i=1..n} 1/prime(i)]. A061015 Numerator of Sum_{i=1..n} 1/prime(i)^2.

LINKS

Table of n, a(n) for n=1..11.

FORMULA

a(n) = Numerator of Sum_{i=1..n} 1/A000040(i)^3.

EXAMPLE

1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.

CROSSREFS

Cf. A000040, A075986/A075987, A106830/A034386.

KEYWORD

easy,frac,nonn

AUTHOR

Jonathan Vos Post, Mar 14 2006

STATUS

approved

A241189

Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

+10
5

1, 7, 11, 127, 1693, 29243, 561623, 13019431, 379503437, 11809225121, 438235268123, 18007758091069, 775817745542929, 36524284093223105, 1938403609207158571, 2160165866032831207, 131893095784520401909, 8844093116997411126541, 628373208972323386101329, 45900898298568589325230523

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

OFFSET

1,2

COMMENTS

a(371) has 1002 decimal digits. - Michael De Vlieger, Jan 27 2016

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..370

EXAMPLE

1/6, 7/30, 11/42, 127/462, 1693/6006, 29243/102102, 561623/1939938, 13019431/44618574, 379503437/1293938646, 11809225121/40112098026, 438235268123/1484147626962, ...

MAPLE

g:= n-> add(1/(ithprime(i)*ithprime(i+1)), i=1..n);

t1:=[seq(g(n), n=1..20)];

t1a:=map(numer, t1); # A241189

t1b:=map(denom, t1); # A241190

MATHEMATICA

Table[Numerator@ Sum[1/(Prime[i + 1] Prime@ i), {i, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)

Accumulate[1/#&/@(Times@@@Partition[Prime[Range[25]], 2, 1])]//Numerator (* Harvey P. Dale, Mar 14 2023 *)

CROSSREFS

Cf. A024451/A002110, A241189/A241190, A241191/A241192.

See also A061015/A061742, A115963/A115964.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese.

STATUS

approved

A075987

Numerator(1+1/prime(1)^3+ ... + 1/prime(n)^3) where prime(k) is the k-th prime.

+10
4

1, 9, 251, 31591, 10862713, 14467532003, 31797494201591, 156248170093443583, 1071839248022015186797, 13041980716182955257968099, 318091971114753602661286869511, 9476548712979446302049526230869201

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

OFFSET

0,2

COMMENTS

The sum in the sequence has limit 1.1747626393. The case of reciprocal squares is in A075986.

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+prime(i)^3 if i=j and 1 otherwise. - Alexander Adamchuk, Jul 08 2006

LINKS

Table of n, a(n) for n=0..11.

Simon Plouffe, Plouffe's Inverter

FORMULA

a(0) = 1; a(n) = a(n-1)*prime(n)^3+(prime(1)*...*prime(n-1))^3.

EXAMPLE

a(2) = 251 so a(3) = 251*p(3)^3 + (2*3)^3 = 251*125 + 216 = 31591.

MATHEMATICA

Table[Det[DiagonalMatrix[Table[Prime[i]^3, {i, 1, n}]]+1], {n, 1, 15}] (* Alexander Adamchuk, Jul 08 2006 *)

PROG

(PARI) a(n) = numerator(1 + sum(k=1, n, 1/prime(k)^3)); \\ Michel Marcus, May 31 2022

CROSSREFS

Cf. A061015, A075986.

KEYWORD

nonn

AUTHOR

Zak Seidov, Sep 28 2002

STATUS

approved

A125708

Numbers k such that A115963(k) is prime.

+10
0

3, 5, 9, 43, 150, 300, 516, 1254

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

OFFSET

1,1

COMMENTS

A115963(n) is the numerator of Sum_{k=1..n} 1/prime(k)^3.

a(9) > 5000. - Michael S. Branicky, Aug 04 2024

LINKS

Table of n, a(n) for n=1..8.

MATHEMATICA

f=0; Do[p=Prime[n]; f=f+1/p^3; g=Numerator[f]; If[PrimeQ[g], Print[{n, p, g}]], {n, 1, 50}]

CROSSREFS

Cf. A115963, A024451, A092062, A061015, A109628.

KEYWORD

hard,more,nonn

AUTHOR

Alexander Adamchuk, Feb 01 2007, Mar 01 2007

EXTENSIONS

a(6) from Alexander Adamchuk, Sep 16 2010

a(7)-a(8) from Amiram Eldar, Feb 18 2019

STATUS

approved

A126225

Least number k > 0 such that the numerator of Sum_{i=1..k} 1/prime(i)^n is a prime.

+10
0

2, 2, 3, 2, 3, 5, 3, 11, 3, 22

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

OFFSET

1,1

COMMENTS

a(12) > 80, a(13) = 30, a(14) = 16, a(18) = 7, a(19) = 3. - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

a(11) > 200, a(12) > 200. - Michel Marcus, May 27 2019

If they exist, a(11) > 1263; a(17) > 954; a(22) > 795; a(23) > 720; a(25) > 570; a(12) = 799, a(15) = 313, a(16) = 780, a(20) = 433, a(21) = 7, a(24) = 4, a(27) = 12, a(29) = 37. - J.W.L. (Jan) Eerland, Jan 26 2023

LINKS

Table of n, a(n) for n=1..10.

EXAMPLE

a(1) = 2 corresponds to A024451(2) = 5, a prime.

a(2) = 2 corresponds to A061015(2) = 13, a prime.

MATHEMATICA

a[n_] := Block[{i = 1, sum = 0}, While[True, sum += 1/Prime[i]^n; If[PrimeQ[Numerator@sum], Return[i]]; i++ ]] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

Table[y[x_, y_]:=Numerator[FullSimplify[Sum[1/Prime[m]^x, {m, 1, y}]]]; k=1; Monitor[Parallelize[While[True, If[PrimeQ[y[n, k]], Break[]]; k++]; k], k], {n, 1, 10}] (* J.W.L. (Jan) Eerland, Jan 25 2023 *)

PROG

(PARI) a(n) = {my(k=1, s=1/prime(k)^n); while (! isprime(numerator(s)), k++; s += 1/prime(k)^n); k; } \\ Michel Marcus, May 27 2019

CROSSREFS

Cf. A024451 (1/p), A061015 (1/p^2), A115963 (1/p^3).

KEYWORD

hard,more,nonn

AUTHOR

Alexander Adamchuk, Mar 08 2007

STATUS

approved

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