A070960 -id:A070960

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A245334

A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

+10
25

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

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

OFFSET

0,2

COMMENTS

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);

A111063(n+1) = sum of n-th row;

T(2*n,n) = A002690(n), central terms;

T(n,0) = n + 1;

T(n,1) = A000290(n), n > 0;

T(n,2) = A011379(n-1), n > 1;

T(n,3) = A047927(n), n > 2;

T(n,4) = A192849(n-1), n > 3;

T(n,5) = A000142(5) * A027810(n-5), n > 4;

T(n,6) = A000142(6) * A027818(n-6), n > 5;

T(n,7) = A000142(7) * A056001(n-7), n > 6;

T(n,8) = A000142(8) * A056003(n-8), n > 7;

T(n,9) = A000142(9) * A056114(n-9), n > 8;

T(n,n-10) = 11 * A051431(n-10), n > 9;

T(n,n-9) = 10 * A049398(n-9), n > 8;

T(n,n-8) = 9 * A049389(n-8), n > 7;

T(n,n-7) = 8 * A049388(n-7), n > 6;

T(n,n-6) = 7 * A001730(n), n > 5;

T(n,n-5) = 6 * A001725(n), n > 5;

T(n,n-4) = 5 * A001720(n), n > 4;

T(n,n-3) = 4 * A001715(n), n > 2;

T(n,n-2) = A070960(n), n > 1;

T(n,n-1) = A052849(n), n > 0;

T(n,n) = A000142(n);

T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

FORMULA

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

EXAMPLE

. 0: 1;

. 1: 2, 1;

. 2: 3, 4, 2;

. 3: 4, 9, 12, 6;

. 4: 5, 16, 36, 48, 24;

. 5: 6, 25, 80, 180, 240, 120;

. 6: 7, 36, 150, 480, 1080, 1440, 720;

. 7: 8, 49, 252, 1050, 3360, 7560, 10080, 5040;

. 8: 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320;

. 9: 10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

MATHEMATICA

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

PROG

(Haskell)

a245334 n k = a245334_tabl !! n !! k

a245334_row n = a245334_tabl !! n

a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

CROSSREFS

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).

Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.

Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.

Cf. A007318, A137948.

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, Aug 30 2014

STATUS

approved

A048249

Number of distinct values produced from sums and products of n unity arguments.

+10
20

1, 2, 3, 4, 6, 9, 11, 17, 23, 30, 44, 60, 80, 114, 156, 212, 296, 404, 556, 770, 1065, 1463, 2032, 2795, 3889, 5364, 7422, 10300, 14229, 19722, 27391, 37892, 52599, 73075, 101301, 140588, 195405, 271024, 376608, 523518, 726812, 1010576, 1405013, 1952498

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

OFFSET

1,2

COMMENTS

Values listed calculated by exhaustive search algorithm.

For n+1 operands (n operations) there are (2n)!/((n!)((n+1)!)) possible postfix forms over a single operator. For each such form, there are 2^n ways to assign 2 operators (here, sum and product). Calculate results and eliminate duplicates.

Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of an integer partition until only one part remains, starting with 1^n. - Gus Wiseman, Sep 29 2018

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..75

Index to sequences related to the complexity of n

Index entries for similar sequences

FORMULA

Equals partial sum of "number of numbers of complexity n" (A005421). - Jonathan Vos Post, Apr 07 2006

EXAMPLE

a(3)=3 since (in postfix): 111** = 11*1* = 1, 111*+ = 11*1+ = 111+* = 11+1* = 2 and 111++ = 11+1+ = 3. Note that at n=7, the 11 possible values produced are the set {1,2,3,4,5,6,7,8,9,10,12}. This is the first n for which there are "skipped" values in the set.

MAPLE

b:= proc(n) option remember; `if`(n=1, {1}, {seq(seq(seq(

[f+g, f*g][], g=b(n-i)), f=b(i)), i=1..iquo(n, 2))})

end:

a:= n-> nops(b(n)):

seq(a(n), n=1..35); # Alois P. Heinz, May 05 2019

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

Table[Length[Select[ReplaceListRepeated[{Array[1&, n]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]], {n, 10}] (* Gus Wiseman, Sep 29 2018 *)

PROG

(Python)

from functools import cache

@cache

def f(m):

if m == 1: return {1}

out = set()

for j in range(1, m//2+1):

for x in f(j):

for y in f(m-j):

out.update([x + y, x * y])

return out

def a(n): return len(f(n))

print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Aug 03 2022

CROSSREFS

Cf. A000792, A005520, A066739, A070960, A201163, A319850, A318949, A319855, A319856, A319909, A319910, A319911.

KEYWORD

nonn,nice

AUTHOR

Tony Bartoletti

EXTENSIONS

More terms from David W. Wilson, Oct 10 2001

a(43)-a(44) from Alois P. Heinz, May 05 2019

STATUS

approved

A319850

Number of distinct positive integers that can be obtained, starting with the initial interval partition (1, ..., n), by iteratively adding or multiplying together parts until only one part remains.

+10
11

1, 2, 5, 21, 94, 446, 2287, 12568, 78509

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..9.

EXAMPLE

The n-th row lists all integers that can be obtained starting with (1, ..., n):

2 3

5 6 7 8 9

9 10 11 12 13 14 15 16 17 18 19 20 21 24 25 26 27 28 30 32 36

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

Table[Length[Select[ReplaceListRepeated[{Range[n]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]], {n, 6}]

CROSSREFS

Cf. A000041, A001055, A001970, A048249, A066739, A066815, A070960, A201163, A318948, A318949, A319841.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 29 2018

STATUS

approved

A319909

Number of distinct positive integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with 1^n.

+10
8

0, 1, 1, 1, 1, 2, 4, 5, 10, 15, 21, 34, 49, 68, 101, 142, 197, 280, 387, 538, 751, 1045, 1442, 2010, 2772, 3865, 5339, 7396, 10273, 14201, 19693

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

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..30.

EXAMPLE

We have

7 = 1+1+1+1+1+1+1,

8 = (1+1)*(1+1+1)+1+1,

9 = (1+1)*(1+1)*(1+1)+1,

10 = (1+1+1+1+1)*(1+1),

12 = (1+1+1)*(1+1+1+1),

so a(7) = 5.

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

mexos[ptn_]:=If[Length[ptn]==0, {0}, Union@@Select[ReplaceListRepeated[{Sort[ptn]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_?(#>1&), mie___, y_?(#>1&), afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]];

Table[Length[mexos[Table[1, {n}]]], {n, 30}]

CROSSREFS

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A070960, A201163, A275870, A319850, A318948, A318949, A319912, A319913.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 01 2018

STATUS

approved

A319855

Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

+10
5

0, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 5, 1, 7, 4, 8, 3, 6, 5, 9, 2, 6, 6, 6, 4, 10, 5, 11, 1, 7, 7, 7, 4, 12, 8, 8, 3, 13, 6, 14, 5, 7, 9, 15, 2, 8, 6, 9, 6, 16, 6, 8, 4, 10, 10, 17, 5, 18, 11, 8, 1, 9, 7, 19, 7, 11, 7, 20, 4, 21, 12, 8, 8, 9, 8, 22, 3, 8

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

OFFSET

1,3

COMMENTS

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

LINKS

Table of n, a(n) for n=1..81.

FORMULA

a(1) = 0, a(n) = max(A056239(n) - A007814(n), 1). - Charlie Neder, Oct 03 2018

EXAMPLE

a(30) = 5 because the minimum number that can be obtained starting with (3,2,1) is 3+2*1 = 5.

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

nexos[ptn_]:=If[Length[ptn]==0, {0}, Union@@Select[ReplaceListRepeated[{Sort[ptn]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]];

Table[Min[nexos[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]], {n, 100}]

CROSSREFS

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319856.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 29 2018

STATUS

approved

A319856

Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

+10
5

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 7, 6, 8, 6, 8, 6, 9, 6, 9, 7, 8, 8, 10, 9, 11, 6, 10, 8, 12, 9, 12, 9, 12, 9, 13, 12, 14, 10, 12, 10, 15, 9, 16, 12, 14, 12, 16, 12, 15, 12, 16, 11, 17, 12, 18, 12, 16, 9, 18, 15, 19, 14, 18, 16, 20, 12, 21, 13

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

OFFSET

1,3

COMMENTS

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

LINKS

Table of n, a(n) for n=1..74.

EXAMPLE

a(30) = 9 because the maximum number that can be obtained starting with (3,2,1) is 3*(2+1) = 9.

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

Table[Max[nexos[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]], {n, 100}]

CROSSREFS

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319855.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 29 2018

STATUS

approved

A071107

a(n) is the greatest integer that can be obtained from the integers {1, 2, 3, ..., n} using each number at most once and the operators +,-,*,/,^.

+10
4

1, 3, 27, 115792089237316195423570985008687907853269984665640564039457584007913129639936

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

OFFSET

1,2

COMMENTS

a(4) = 2^(4^4) = 2^256 = 115792089237316195423570985008687907853269984665640564039457584007913129639936 with 78 digits. a(5) = 2^(3^(4^6)) with more than 10^1950 digits.

LINKS

Table of n, a(n) for n=1..4.

EXAMPLE

a(3) = 27 because 3^(2+1) = 27 is the greatest integer that can be obtained by using 1, 2, 3 once and the operations +, -, *, /, ^.

CROSSREFS

Cf. A070960.

KEYWORD

nonn

AUTHOR

Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002

STATUS

approved

A358805

Numbers k such that k! + (k!/2) + 1 is prime.

+10
4

4, 5, 7, 11, 12, 14, 18, 28, 30, 62, 135, 153, 275, 584, 630, 1424, 1493, 4419, 8492, 10950

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

OFFSET

1,1

COMMENTS

Numbers k such that A070960(k)+1 is prime.

No more terms < 10000. - Vaclav Kotesovec, Dec 12 2022

LINKS

Table of n, a(n) for n = 1..19

Arsen Vardanyan, A list of prime numbers

PROG

(PARI) is(k) = isprime(k!+(k!/2)+1);

CROSSREFS

Cf. A070960, A358878.

KEYWORD

nonn,more

AUTHOR

Arsen Vardanyan, Dec 01 2022

EXTENSIONS

a(18)-a(19) from Vaclav Kotesovec, Dec 09 2022

a(20) from Michael S. Branicky, Aug 02 2024

STATUS

approved

A358878

Numbers k such that k! + (k!/2) - 1 is prime.

+10
3

2, 5, 7, 15, 20, 47, 84, 138, 169, 257, 263, 431, 559, 2939, 4403, 4870, 5273

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

OFFSET

1,1

COMMENTS

Numbers k such that A070960(k) - 1 is prime.

No more terms < 10000. - Vaclav Kotesovec, Dec 12 2022

LINKS

Table of n, a(n) for n=1..17.

PROG

(PARI) is(k) = isprime(k!+(k!/2)-1);

CROSSREFS

Cf. A070960, A358805.

KEYWORD

nonn,more

AUTHOR

Arsen Vardanyan, Dec 04 2022

EXTENSIONS

a(14)-a(17) from Vaclav Kotesovec, Dec 07 2022

STATUS

approved

A071603

Number of different positive integers that we can obtain from the integers {1,2,...,n} using each number at most once and the operators +, -, *, /, where intermediate subexpressions must be integers.

+10
0

1, 3, 9, 31, 121, 542, 2868, 16329, 106762, 758155, 6142570

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

OFFSET

1,2

LINKS

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

Index entries for similar sequences

EXAMPLE

a(4)=31 because we can obtain the positive integers 1,2,...,28 and 30,32,36 by using the integers {1, 2, 3, 4} at most once and the four operations. For example 30 = 3*2*(4+1).

PROG

(Python)

def a(n):

R = dict() # index of each reachable subset is [card(s)-1][s]

for i in range(n): R[i] = dict()

for i in range(1, n+1): R[0][(i, )] = {i}

reach = set(range(1, n+1))

for j in range(1, n):

for i in range((j+1)//2):

for s1 in R[i]:

for s2 in R[j-1-i]:

if set(s1) & set(s2) == set():

s12 = tuple(sorted(set(s1) | set(s2)))

if s12 not in R[len(s12)-1]:

R[len(s12)-1][s12] = set()

for a in R[i][s1]:

for b in R[j-1-i][s2]:

allowed = [a+b, a*b, a-b, b-a]

if a!=0 and b%a==0: allowed.append(b//a)

if b!=0 and a%b==0: allowed.append(a//b)

R[len(s12)-1][s12].update(allowed)

reach.update(allowed)

return len(set(r for r in reach if r > 0 and r.denominator == 1))

print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 01 2022

CROSSREFS

Cf. A060315, A070960, A071848.

KEYWORD

more,nonn

AUTHOR

Koksal Karakus (karakusk(AT)hotmail.com), Jun 02 2002

EXTENSIONS

a(10)-a(11) from Michael S. Branicky, Jul 01 2022

STATUS

approved

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