A046063 -id:A046063 - cp's OEIS frontend

A114168 Numbers n such that p(8n) is prime, where p(n) is the number of partitions of n.

21, 27, 55, 162, 267, 321, 364, 396, 557, 1056, 1114, 1385, 1684, 1880, 2031, 3825, 4069, 4155, 4337, 4561, 7721, 7816, 8179, 8452, 9962, 15885, 16871, 17024, 17040, 17670, 22186, 23037, 26782, 31307, 35364, 35442, 38430, 42307

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..630

Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.

Select[ Range[25000], PrimeQ[PartitionsP[8# ]] &]

is(n)=isprime(numbpart(8*n)) \\ Charles R Greathouse IV, Feb 17 2017

A114169 Numbers n such that p(9n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

4, 24, 144, 277, 278, 303, 319, 352, 518, 2279, 2405, 2578, 3400, 3787, 4273, 4457, 7603, 9145, 9858, 10774, 10988, 11545, 12954, 14120, 14674, 17537, 18193, 18602, 18919, 21955, 29775, 30559, 31504, 34160, 35618, 35655

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..555

Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.

Select[ Range[22222], PrimeQ[PartitionsP[9# ]] &]

is(n)=isprime(numbpart(9*n)) \\ Charles R Greathouse IV, Feb 17 2017

A114170 Numbers n such that p(10n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

44, 1108, 1302, 1504, 1829, 1847, 2267, 2537, 3060, 3289, 3324, 3997, 4138, 6175, 6505, 7266, 9733, 10177, 11483, 12708, 12881, 13632, 14136, 14414, 15917, 16409, 17614, 19133, 19381, 21966, 22967, 30565, 30744, 31655, 33783

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..607

Cf. A000041, A046063, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170, A070177.

Select[ Range[20000], PrimeQ[PartitionsP[10# ]] &]

is(n)=isprime(numbpart(10*n)) \\ Charles R Greathouse IV, Feb 17 2017

A284594 Numbers whose square has a prime number of partitions.

Original entry on oeis.org

2, 6, 29, 36, 2480, 14881

Offset: 1

Serge Batalov, Mar 29 2017

Because asymptotically A072213(n) = A000041(n^2) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2), the sum of the prime probabilities ~ 1/log(A072213(n)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
Curiously, both A000041(6^2) and A000041(6^4) are prime; in addition, A000041(6^3) and A000041(6^1) are prime, but for no other powers A000041(6^k) is known (or can be expected) to be prime.
a(7) > 649350.

			a(2) = 6 is in the sequence because A000041(6^2) = 17977 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A285086, A285087, A285088.

for(n=1,2500,if(ispseudoprime(numbpart(n^2)),print1(n,", ")))

A285086 Numbers n such that the number of partitions of n^2+1 (=A000041(n^2+1)) is prime.

Original entry on oeis.org

1, 2, 3914

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n^2+1) ~ exp(Pi*sqrt(2/3*(n^2+1))) / (4*sqrt(3)*(n^2+1)), the sum of the prime probabilities ~ 1/log(A000041(n^2+1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(4) > 90000.

			a(2) = 2 is in the sequence because A000041(2^2+1) = 7 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285087, A285088.

for(n=1,3920,if(ispseudoprime(numbpart(n^2+1)),print1(n,", ")))

A285087 Numbers n such that the number of partitions of n^2-1 is prime.

Original entry on oeis.org

2, 13, 21, 46909

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n^2-1) ~ exp(Pi*sqrt(2/3*(n^2-1))) / (4*sqrt(3)*(n^2-1)), the sum of the prime probabilities ~1/log(A000041(n^2-1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(5) > 50000.

			13 is in the sequence because A000041(13^2-1) = 228204732751 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285086, A285088.

for(n=1,2000,if(ispseudoprime(numbpart(n^2-1)),print1(n,", ")))

from itertools import count, islice
from sympy import isprime, npartitions
def A285087_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: isprime(npartitions(n**2-1)), count(max(startvalue,1)))
A285087_list = list(islice(A285087_gen(),3)) # Chai Wah Wu, Nov 20 2023

{n: A000041(n^2-1) in A000040}.

A285088 Numbers n such that the number of partitions of n(n+1)/2 (=A000041(A000217(n))) is prime.

Original entry on oeis.org

2, 3, 8, 3947, 43968, 61681

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n*(n+1)/2) ~ exp(Pi*sqrt(2/3*(n*(n+1)/2))) / (4*sqrt(3)*(n*(n+1)/2)), the sum of the prime probabilities ~1/log(A000041(n*(n+1)/2)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

			a(3) = 8 is in the sequence because A000041(8*9/2) = 17977 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285086, A285087.

for(n=1,2000,if(ispseudoprime(numbpart(n*(n+1)/2)),print1(n,", ")))

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

cp's OEIS Frontend

A114168 Numbers n such that p(8n) is prime, where p(n) is the number of partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A114169 Numbers n such that p(9n) is prime, where p(n) is the number of partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A114170 Numbers n such that p(10n) is prime, where p(n) is the number of partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A284594 Numbers whose square has a prime number of partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A285086 Numbers n such that the number of partitions of n^2+1 (=A000041(n^2+1)) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A285087 Numbers n such that the number of partitions of n^2-1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A285088 Numbers n such that the number of partitions of n(n+1)/2 (=A000041(A000217(n))) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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