A051143 -id:A051143 - cp's OEIS frontend

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

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

A049575 Prime partition numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939

Offset: 1

G. L. Honaker, Jr.

Alois P. Heinz, Table of n, a(n) for n = 1..100
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 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.

Intersection of A000040 and A000041.
Cf. A046063, A051143.
Cf. A038753, A065728. - Reinhard Zumkeller, Nov 03 2009

lst={};Do[a=PartitionsP[n];If[PrimeQ[a],AppendTo[lst,a]],{n,2*6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
Select[PartitionsP[Range[1000]],PrimeQ] (* Harvey P. Dale, Mar 11 2013 *)

for(n=0,10^3,my(p=numbpart(n));if(isprime(p),print1(p,", "))); \\ Joerg Arndt, May 09 2013

a(n) = A000041(A046063(n)) = A000040(A051143(n)). - M. F. Hasler, Oct 19 2008

A010051(a(n))*A167392(a(n)) = 1. - Reinhard Zumkeller, Nov 03 2009

More terms from James Sellers and Christian G. Bower, Oct 15 1999.

A050811 Partition numbers rounded to nearest integer given by the Hardy-Ramanujan approximate formula.

Original entry on oeis.org

2, 3, 4, 6, 9, 13, 18, 26, 35, 48, 65, 87, 115, 152, 199, 258, 333, 427, 545, 692, 875, 1102, 1381, 1725, 2145, 2659, 3285, 4046, 4967, 6080, 7423, 9037, 10974, 13293, 16065, 19370, 23304, 27977, 33519, 40080, 47833, 56981, 67757, 80431, 95316

Offset: 1

Patrick De Geest, Oct 15 1999

The mounting error seems to be approximately A035949(n-3), n >= 4. - Alonso del Arte, Jul 28 2011

This conjecture is false, for correct approximation see the formula below. - Vaclav Kotesovec, Apr 03 2017

John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 95.

Dr. Math, Partitioning the Integers
Dr. Math, Partitioning an Integer
D. Rusin, Additive Partitions of Number
F. Ruskey, Generate Numerical Partitions
Eric Weisstein's World of Mathematics, Partition Function P
OEIS Wiki, Partition function

Cf. A000041, A035949, A049575, A051143.

A050811:=n->round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))): seq(A050811(n), n=1..70); # Wesley Ivan Hurt, Sep 11 2015

f[n_] := Round[ E^(Sqrt[2n/3] Pi)/(4Sqrt[3] n)]; Array[f, 45] (* Alonso del Arte, May 21 2011, corrected by Robert G. Wilson v, Sep 11 2015 *)

a(n)=round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))) \\ Charles R Greathouse IV, May 01 2012

input N:print round(#e^(pi(1)*sqrt(2*N/3))/(4*N*sqrt(3)))

a(n) = round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))). - Alonso del Arte, May 21 2011

a(n) - A000041(n) ~ (1/Pi + Pi/72) * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (9 + Pi^2/48)*Pi/((72 + Pi^2)*sqrt(6*n))). - Vaclav Kotesovec, Apr 03 2017

a(1) = 1 replaced by 2, a(2) = 2 replaced by 3. - Alonso del Arte, D. S. McNeil, Aug 07 2011

A091689 Smallest partition number with n-th prime as factor.

Original entry on oeis.org

2, 3, 5, 7, 11, 3718, 386155, 627, 8349, 2436, 75175, 34262962, 14883, 3010, 526823, 281589, 386155, 1064144451, 124754, 63261, 105558, 2552338241, 4565, 1958, 75175, 101, 12132164, 118114304, 37274405776748077, 1505499, 37338, 6185689, 2323520, 966467, 90175434980549623

Offset: 1

Reinhard Zumkeller, Jan 29 2004

nonn

Erdős conjectured that every prime divides at least one value of the partition function, see Ahlgren and Ono link.

			For n = 10, A000040(10) = 29: a(10) = A000041(26) = 2436 = 29*7*3*2*2, as 29 does not divide smaller partition numbers.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Scott Ahlgren and Ken Ono, Addition and Counting: The Arithmetic of Partitions, Notices of the AMS, 48 (2001) pp. 978-984. See p. 982.
Gerard P. Michon, Table of partition function p(n) (n=0 through 4096).
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A051143, A049575, A046063.

Cf. A000041, A091690.

a(n) = A000041(A091690(n)).

More terms from Amiram Eldar, May 16 2025

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

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A049575 Prime partition numbers.

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A050811 Partition numbers rounded to nearest integer given by the Hardy-Ramanujan approximate formula.

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A091689 Smallest partition number with n-th prime as factor.

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