A035359 -id:A035359 - cp's OEIS frontend

A051005 PartitionsQ[ A035359 ], i.e., prime values of PartitionsQ.

2, 2, 3, 5, 89, 29927, 444793, 602644050950309, 5907806880101973271193081, 442874899733097781915111718440153

Offset: 1

The next term is known (see A035359) but is too big to include here.

Eric Weisstein's World of Mathematics, Partition Function Q Congruences.
Eric Weisstein's World of Mathematics, Partition Function Q

Reap[For[n = 1, n <= 2500, n++, If[PrimeQ[q = PartitionsQ[n]], Print["n = ", n, " Q = ", q]; Sow[q]]]][[2, 1]] (* Jean-François Alcover, Oct 19 2012 *)
Select[PartitionsQ[Range[2500]],PrimeQ] (* Harvey P. Dale, Aug 11 2015 *)

A046063 Numbers k such that the k-th partition number A000041(k) is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, 188, 212, 216, 302, 366, 417, 440, 491, 498, 525, 546, 658, 735, 753, 825, 841, 863, 1085, 1086, 1296, 1477, 1578, 1586, 1621, 1793, 2051, 2136, 2493, 2502, 2508, 2568, 2633, 2727, 2732, 2871, 2912, 3027, 3098, 3168, 3342, 3542, 3641, 4118

Offset: 1

Eric W. Weisstein

The corresponding primes are given in A049575. - Joerg Arndt, May 09 2013

Max Alekseyev, Table of n, a(n) for n = 1..4967 (contains all terms below 10^8)
Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
G. P. Michon, Table of partition function p(n) (n=0 through 4096)
G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), Volume 1(6) (2014), ISSN: 2349-8862.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Eric Weisstein's World of Mathematics, Partition Function P.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A000041, A035359, A049575, A051143, A111036, A111045, A114165, A111389, A113499, A114166, A114167, A114168, A114169, A114170, A115214.

Select[ Range@3341, PrimeQ@ PartitionsP@# &] (* Robert G. Wilson v *)

for(n=0,10^5,my(p=numbpart(n));if(isprime(p),print1(n,", "))); \\ Joerg Arndt, May 09 2013

from sympy import isprime, npartitions
print([n for n in range(1, 5001) if isprime(npartitions(n))]) # Indranil Ghosh, Apr 10 2017

b-file extended by Max Alekseyev, Jul 07 2009, Jun 14 2011, Jan 08 2012, May 19 2014

A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

In short, A001055(a(n)) belongs to A000040.

			Factorizations of selected terms:
  (4)    (8)      (16)       (24)       (60)       (96)
  (2*2)  (2*4)    (2*8)      (3*8)      (2*30)     (2*48)
         (2*2*2)  (4*4)      (4*6)      (3*20)     (3*32)
                  (2*2*4)    (2*12)     (4*15)     (4*24)
                  (2*2*2*2)  (2*2*6)    (5*12)     (6*16)
                             (2*3*4)    (6*10)     (8*12)
                             (2*2*2*3)  (2*5*6)    (2*6*8)
                                        (3*4*5)    (3*4*8)
                                        (2*2*15)   (4*4*6)
                                        (2*3*10)   (2*2*24)
                                        (2*2*3*5)  (2*3*16)
                                                   (2*4*12)
                                                   (2*2*3*8)
                                                   (2*2*4*6)
                                                   (2*3*4*4)
                                                   (2*2*2*12)
                                                   (2*2*2*2*6)
                                                   (2*2*2*3*4)
                                                   (2*2*2*2*2*3)

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Cf. A001222, A033833, A045783, A181819, A181821, A326622, A330972, A330973, A330993.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[facs[#]]]&]

A330992 Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.

Original entry on oeis.org

4, 8, 16, 24, 60, 0, 0, 96, 0, 144, 216, 0, 0, 0, 288, 0, 0, 0, 768, 0, 0, 0, 0, 0, 864, 8192, 0, 0, 1080, 0, 0, 0, 1800, 3072, 0, 0, 0, 0, 0, 0, 0, 2304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3456, 0, 3600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24576

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

			Factorizations of the initial positive terms are:
  4    8      16       24       60       96
  2*2  2*4    2*8      3*8      2*30     2*48
       2*2*2  4*4      4*6      3*20     3*32
              2*2*4    2*12     4*15     4*24
              2*2*2*2  2*2*6    5*12     6*16
                       2*3*4    6*10     8*12
                       2*2*2*3  2*5*6    2*6*8
                                3*4*5    3*4*8
                                2*2*15   4*4*6
                                2*3*10   2*2*24
                                2*2*3*5  2*3*16
                                         2*4*12
                                         2*2*3*8
                                         2*2*4*6
                                         2*3*4*4
                                         2*2*2*12
                                         2*2*2*2*6
                                         2*2*2*3*4
                                         2*2*2*2*2*3

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

All positive terms belong to A025487 and also A033833.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of partitions is prime are A046063.
Numbers whose number of strict partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers with a prime number of factorizations are A330991.
The least number with exactly 2^n factorizations is A330989(n).
Cf. A001222, A045783, A325238, A330972, A330973, A330976, A330993, A330998.

More terms from Jinyuan Wang, Jul 07 2021

A331022 Numbers k such that the number of strict integer partitions of k is a power of 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 16, 20, 29, 34, 45

Offset: 0

Gus Wiseman, Jan 10 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.
Conjecture: This sequence is finite.
Conjecture: The analogous sequence for non-strict partitions is: 0, 1, 2.
Next term > 5*10^4 if it exists. - Seiichi Manyama, Jan 12 2020

			The strict integer partitions of the initial terms:
  (1)  (2)  (3)    (4)    (6)      (9)
            (2,1)  (3,1)  (4,2)    (5,4)
                          (5,1)    (6,3)
                          (3,2,1)  (7,2)
                                   (8,1)
                                   (4,3,2)
                                   (5,3,1)
                                   (6,2,1)

The version for primes instead of powers of 2 is A035359.
The version for factorizations instead of strict partitions is A330977.
Numbers whose number of partitions is prime are A046063.
Cf. A000009, A000079, A001055, A036498, A045778, A318286, A330989, A330990, A330994/A330995, A330996.

Select[Range[0,1000],IntegerQ[Log[2,PartitionsQ[#]]]&]

A265835 Numbers n such that A015128(n)/2 is prime.

Original entry on oeis.org

2, 4, 16, 36, 400, 1296, 1521, 52441

Offset: 1

Vaclav Kotesovec, Dec 16 2015

Next term, if it exists, is greater than 4000000. - Vaclav Kotesovec, updated Apr 12 2017

The values of a(n) are the squares of these integers for 1 < n < 9: 2, 4, 6, 20, 36, 39, 229. Squares also appear in the sequence of numbers k such that A015128(k)/2 is semiprime. - Altug Alkan, Dec 16 2015

			4 is a term because A015128(4)/2 = 14/2 = 7 is prime.

Cf. A014968, A015128, A035359, A046063, A299961.

Select[Range[2000], PrimeQ[Sum[PartitionsP[#-k]*PartitionsQ[k], {k, 0, #}]/2] &]

a015128(n) = polcoeff(exp(sum(m=1, n\2+1, 2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/(2*m-1))), n);
for(n=1, 1e3, if(ispseudoprime(a015128(n)/2), print1(n, ", "))) \\ Altug Alkan, Dec 16 2015

A330994 Numerator of P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 3, 11, 15, 21, 14, 77, 101, 135, 176, 231, 297, 385, 245, 627, 198, 1002, 1255, 1575, 979, 812, 1505, 1859, 4565, 1401, 3421, 2783, 1449, 6155, 4961, 17977, 21637, 26015, 31185, 1778, 2123, 26587, 63261, 75175, 44567, 17593, 8911, 49091

Offset: 0

Gus Wiseman, Jan 08 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.

The denominators are A330995.
The rounded quotients are A330996.
The same for factorizations is A331023.
Cf. A000009, A000041, A003238, A005117, A035359, A046063, A331022.

Table[PartitionsP[n]/PartitionsQ[n],{n,0,100}]//Numerator

A330994/A330995 = A000041/A000009.

A330995 Denominator P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 1, 3, 4, 5, 3, 15, 18, 22, 27, 32, 38, 46, 27, 64, 19, 89, 104, 122, 71, 55, 96, 111, 256, 74, 170, 130, 64, 256, 195, 668, 760, 864, 982, 53, 60, 713, 1610, 1816, 1024, 384, 185, 970, 3264, 1829, 4097, 4582, 5120, 5718, 3189, 7108, 2639

Offset: 0

Gus Wiseman, Jan 08 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.

Conjecture: The only 1's occur at n = 0, 1, 2, 7.

The numerators are A330994.
The rounded quotients are A330996.
The same for factorizations is A331024.
Cf. A000009, A000041, A001055, A003238, A005117, A035359, A045778, A046063, A331022.

Table[PartitionsP[n]/PartitionsQ[n],{n,0,100}]//Denominator

A330994/A330995 = A000041/A000009.

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Gus Wiseman, Jan 07 2020

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

The same for powers of 2 (instead of primes) is A330990.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Factorizations of a number's inverse prime shadow are A318284.
Numbers with a prime number of factorizations are A330991.
Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

A001055(A181821(a(n))) belongs to A000040.

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[strfacs[#]]]&]

cp's OEIS Frontend

A051005 PartitionsQ[ A035359 ], i.e., prime values of PartitionsQ.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A046063 Numbers k such that the k-th partition number A000041(k) is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A330992 Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A331022 Numbers k such that the number of strict integer partitions of k is a power of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A265835 Numbers n such that A015128(n)/2 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A330994 Numerator of P(n)/Q(n) = A000041(n)/A000009(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A330995 Denominator P(n)/Q(n) = A000041(n)/A000009(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

Views

Author