A049575 -id:A049575 - 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

A234470 Number of ways to write n = k + m with k > 0 and m > 2 such that p(k + phi(m)/2) is prime, where p(.) is the partition function (A000041) and phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 5, 4, 4, 4, 2, 2, 3, 5, 4, 2, 4, 2, 3, 2, 3, 2, 3, 1, 0, 3, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 3, 3, 3, 1, 0, 4, 2, 4, 1, 1, 2, 2, 3, 2, 2, 0, 2, 2, 1, 2, 2, 1, 1, 2, 2, 4, 2, 1, 0, 1, 3, 1, 0, 2, 4, 3, 1, 6, 2, 2, 1, 2, 4, 3, 1, 2, 6, 2, 3, 2, 2, 2, 2, 3, 3

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

Conjecture: a(n) > 0 if n > 3 is not among 27, 34, 50, 61, 74, 78, 115, 120, 123, 127.

This implies that there are infinitely many primes in the range of the partition function p(n).

			a(26) = 1 since 26 = 2 + 24 with p(2 + phi(24)/2) = p(6) = 11 prime.
a(54) = 1 since 54 = 27 + 27 with p(27 + phi(27)/2) = p(36) = 17977 prime.
a(73) = 1 since 73 = 1 + 72 with p(1 + phi(72)/2) = p(36) = 17977 prime.
a(110) = 1 since 110 = 65 + 45 with p(65 + phi(45)/2) = p(77) = 10619863 prime.
a(150) = 1 since 150 = 123 + 27 with p(123 + phi(27)/2) = p(132) = 6620830889 prime.
a(170) = 1 since 170 = 167 + 3 with p(167 + phi(3)/2) = p(168) = 228204732751 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..6500
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A000041, A049575, A232504, A233307, A233346, A233918, A234200, A234246, A234309, A234337, A234344, A234347, A234359, A234360, A234451, A234475

f[n_,k_]:=PartitionsP[k+EulerPhi[n-k]/2]
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A234569 Primes p with P(p-1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

3, 5, 7, 37, 367, 499, 547, 659, 1087, 1297, 1579, 2137, 2503, 3169, 3343, 4457, 4663, 5003, 7459, 9293, 16249, 23203, 34667, 39971, 41381, 56383, 61751, 62987, 72661, 77213, 79697, 98893, 101771, 127081, 136193, 188843, 193811, 259627, 267187, 282913, 315467, 320563, 345923, 354833, 459029, 482837, 496477, 548039, 641419, 647189

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

By the conjecture in A234567, this sequence should have infinitely many terms. It seems that a(n+1) < a(n) + a(n-1) for all n > 5.
The b-file lists all terms not exceeding the 500000th prime 7368787. Note that P(a(113)-1) is a prime having 2999 decimal digits.
See also A234572 for primes of the form P(p-1) with p prime.

			a(1) = 3 since P(2-1) = 1 is not prime, but P(3-1) = 2 is prime.
a(2) = 5 since P(5-1) = 5 is prime.
a(3) = 7 since P(7-1) = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..113
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234572, A234615, A234644

n=0;Do[If[PrimeQ[PartitionsP[Prime[k]-1]],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]

A236103 Number of distinct partition numbers dividing n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 1, 3, 4, 2, 1, 3, 1, 3, 3, 4, 1, 3, 2, 2, 2, 3, 1, 6, 1, 2, 3, 2, 3, 3, 1, 2, 2, 3, 1, 5, 1, 4, 4, 2, 1, 3, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 2, 2, 1, 5, 2, 2, 2, 4, 1, 6

Offset: 1

Omar E. Pol, Jan 21 2014

nonn

			For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20 and three of them are also partition numbers: 1, 2, 5, so a(20) = 3.
For n = 42 the divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and five of them are also partition numbers: 1, 2, 3, 7, 42, so a(42) = 5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000041, A001221, A049575, A078506, A085543, A167392, A236102, A236105, A236107, A236108.

p = {1}; Table[If[n >= Last@p, AppendTo[p, PartitionsP[1 + Length@p]]]; Length@Select[p, Mod[n, #] == 0 &], {n, 90}] (* Giovanni Resta, Jan 22 2014 *)

From Amiram Eldar, Jan 01 2024: (Start)
a(n) = Sum_{d|n} A167392(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A078506 = 2.510597... . (End)

A236108 Nonprimes whose proper divisors are partition numbers.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 303, 505, 707, 1111, 10201, 35954, 53931, 89885, 125839, 197747, 1815677, 21239726, 31859589, 53099315, 74339041, 116818493, 323172529, 1072606163, 13241661778, 19862492667, 33104154445, 46345816223, 72829139779

Offset: 1

Omar E. Pol, Jan 22 2014

nonn

Known terms are squares of A049575 or products of 2 distinct terms of A049575. - Michel Marcus, Jan 25 2023

This conjecture holds for terms <= 10^16. - David A. Corneth, Jan 25 2023

			10 is in the sequence because 10 is a nonprime number and the proper divisors of 10 are 1, 2, 5, which are also partition numbers.

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

Cf. A000041, A018252, A049575, A167392.

Cf. A236102, A236103, A236105, A236107, A236110.

isA000041 := proc(n)
    local k,P;
    for k from 1 do
        P := combinat[numbpart](k) ;
        if P > n then
            return false;
        elif P = n then
            return true ;
        end if;
    end do:
end proc:
isA236108 := proc(n)
    local pdvs,d ;
    if n =1 or isprime(n) then
        return false;
    end if;
    pdvs := numtheory[divisors](n) minus {n} ;
    for d in pdvs do
        if not isA000041(d) then
            return false;
        end if;
    end do:
    return true;
end proc:
for n from 1 to 300000 do
    if isA236108(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 29 2014

partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];
Select[Range[2, 10000],
 If[! PrimeQ[#],
ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* Julien Kluge, Dec 03 2016 *)

a(17)-a(26) from R. J. Mathar, Jan 29 2014
a(27)-a(32) from Jon E. Schoenfield, Feb 05 2014
a(33)-a(34) from Michel Marcus, Jan 24 2023
More terms from David A. Corneth, Jan 25 2023

A234572 Primes of the form P(p-1), where p is a prime and P(.) is the partition function (A000041).

Original entry on oeis.org

2, 5, 11, 17977, 790738119649411319, 2058791472042884901563, 27833079238879849385687, 8121368081058512888507057, 675004412390512738195023734124239, 1398703012615213588677365804960180341, 16193798232344933888778097136641377589301, 204931453786129197483756438132982529754356479553, 3019564607799532159016586951616642980389816614848623, 22757918197082858017617136646280039394687006502870793231847, 1078734573992480956821414895441907729656949308800686938161281

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Though the primes in this sequence are very rare, by the conjecture in A234567 there should be infinitely many such primes.

See A234569 for a list of known primes p with P(p-1) also prime.

			a(1) = 2 since 2 = P(3-1) with 2 and 3 both prime.
a(2) = 5 since 5 = P(5-1) with 5 prime.
a(3) = 11 since 11 = P(7-1) with 7 and 11 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50

Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234569

p[n_]:= A234569(n)
Table[PartitionsP[p[n]-1],{n,1,15}]

a(n) = A000041(A234569(n)-1).

A236102 Numbers whose divisors are partition numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 77, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939

Offset: 1

Omar E. Pol, Jan 21 2014

nonn

By definition all terms are partition numbers.
All members of A049575 are in this sequence.
Conjecture: the only composite numbers in this sequence are 15, 22, and 77. - Jon E. Schoenfield, Feb 05 2014

			15 is in the sequence because the divisors of 15 are 1, 3, 5, 15, which are also partition numbers.

Cf. A000041, A027750, A049575, A236103, A236105, A236107, A236108, A236110, A236111.

More terms from Jon E. Schoenfield, Feb 05 2014

A121062 Partition numbers mod 4.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 3, 3, 2, 2, 2, 0, 1, 1, 3, 0, 3, 1, 1, 2, 3, 0, 2, 3, 3, 2, 0, 2, 2, 1, 0, 2, 1, 3, 2, 3, 1, 1, 3, 1, 2, 3, 2, 1, 3, 2, 2, 2, 1, 1, 2, 3, 1, 3, 3, 0, 3, 2, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 3, 1, 0, 1, 3, 1, 0, 0, 3, 3, 0, 2, 0, 3, 3, 1, 0, 1, 2, 1, 1, 1, 1, 3, 3, 1, 0, 3, 0, 2, 0, 3, 0, 2, 3, 2, 1

Offset: 0

Jonathan Vos Post, Aug 10 2006

P_n==0: 11, 15, 21, 26, 30, 55, 58, 59, 62, 66, 70, 74, 75, 78, 80, 84, 94, 96, 98, 100, ... A237278.
P_n==1: 0, 1, 4, 12, 13, 17, 18, 29, 32, 36, 37, 39, 43, 48, 49, 52, 61, 67, 69, 71, 73, ... A237279.
P_n==2: 2, 8, 9, 10, 19, 22, 25, 27, 28, 31, 34, 40, 42, 45, 46, 47, 50, 57, 64, 65, 79, ... A237280.
P_n==3: 3, 5, 6, 7, 14, 16, 20, 23, 24, 33, 35, 38, 41, 44, 51, 53, 54, 56, 60, 63, 68, ... A237281.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A010873, A049575.

Cf. A237278, A237279, A237280, A237281.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     (b(n, i-1)+b(n-i, min(n-i, i))) mod 4))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..104);  # Alois P. Heinz, Dec 20 2024

f[n_] := Mod[PartitionsP@n, 4]; Table[f@n, {n, 0, 104}] (* Robert G. Wilson v *)

a(n) = numbpart(n) % 4; \\ Michel Marcus, Jun 29 2016

a(n) = A000041(n) mod 4 = A010873(A000041(n)).

a(n) = A000025(n) mod 4. - John M. Campbell, Jun 29 2016

More terms from Robert G. Wilson v, Aug 17 2006

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

A051143 Numbers k such that the k-th prime is a partition number.

Original entry on oeis.org

1, 2, 3, 4, 5, 26, 2061, 702993, 307058572, 3350187739, 9088200428, 43794115173, 51932790219, 378210209388, 521301342188, 297064987225918, 19677201507658441, 437852535314831447, 1673669998972800207, 29252504332047744188, 42842701894337201916

Offset: 1

G. L. Honaker, Jr.

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000040, A000041, A000720, A046063, A049575.

PrimePi@Select[PartitionsP@Range@301, PrimeQ@# &] (* Robert G. Wilson v, Nov 14 2005 *)

a(n) = A000720(A049575(n)).

a(16)-a(21) using Kim Walisch's primecount, from Amiram Eldar, Jul 26 2019

cp's OEIS Frontend

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

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A234470 Number of ways to write n = k + m with k > 0 and m > 2 such that p(k + phi(m)/2) is prime, where p(.) is the partition function (A000041) and phi(.) is Euler's totient function.

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A234569 Primes p with P(p-1) also prime, where P(.) is the partition function (A000041).

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A236103 Number of distinct partition numbers dividing n.

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A236108 Nonprimes whose proper divisors are partition numbers.

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A234572 Primes of the form P(p-1), where p is a prime and P(.) is the partition function (A000041).

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A236102 Numbers whose divisors are partition numbers.

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A121062 Partition numbers mod 4.

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