A046063 -id:A046063 - cp's OEIS frontend

A049575 Prime partition numbers.

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.

A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

Original entry on oeis.org

3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852

Offset: 1

Olivier Gérard

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

			From _Gus Wiseman_, Jan 13 2020: (Start)
Strict partitions of a(1) = 3 through a(4) = 7:
  (3)    (4)    (5)    (7)
  (2,1)  (3,1)  (3,2)  (4,3)
                (4,1)  (5,2)
                       (6,1)
                       (4,2,1)
(End)

Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Partition Function Q
Eric Weisstein's World of Mathematics, Partition Function Q-Congruences

The non-strict version is A046063.
The version for powers of 2 instead of primes is A331022.
The version for factorizations instead of strict partitions is A330991.
The version for strict factorizations instead of strict partitions is A331201.
Cf. A000009, A051005, A056848, A265835.

n = 1; A035359 = {}; While[n < 10^7, n++; If[ PrimeQ[ PartitionsQ[n]], Print[n]; AppendTo[A035359, n]]]; A035359 (* Jean-François Alcover, Oct 12 2011 *)

More terms from Eric W. Weisstein
a(12) from Max Alekseyev, Jul 04 2009
a(13)-a(14) from Giovanni Resta, Jun 05 2015, Jun 11 2015
a(15)-a(17) from Max Alekseyev, Jul 10 2015

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[#]]]&]

A052002 Numbers with an odd number of partitions.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 12, 13, 14, 16, 17, 18, 20, 23, 24, 29, 32, 33, 35, 36, 37, 38, 39, 41, 43, 44, 48, 49, 51, 52, 53, 54, 56, 60, 61, 63, 67, 68, 69, 71, 72, 73, 76, 77, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 95, 99, 102, 104, 105, 107, 111, 114, 115, 118, 119, 121

Offset: 1

Patrick De Geest, Nov 15 1999

A052003(n) = A000041(a(n+1)). - Reinhard Zumkeller, Nov 03 2015

Also, numbers having an odd number of partitions into distinct odd parts; that is, numbers m such that A000700(m) is odd. For example, 16 is in the list since 16 has 5 partitions into distinct odd parts, namely, 1 + 15, 3 + 13, 5 + 11, 7 + 9 and 1 + 3 + 5 + 7. See Formula section for a proof. - Peter Bala, Jan 22 2017

			From _Gus Wiseman_, Jan 13 2020: (Start)
The partitions of the initial terms are:
  (1)  (3)    (4)     (5)      (6)       (7)
       (21)   (22)    (32)     (33)      (43)
       (111)  (31)    (41)     (42)      (52)
              (211)   (221)    (51)      (61)
              (1111)  (311)    (222)     (322)
                      (2111)   (321)     (331)
                      (11111)  (411)     (421)
                               (2211)    (511)
                               (3111)    (2221)
                               (21111)   (3211)
                               (111111)  (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)
(End)

Clark Kimberling, Table of n, a(n) for n = 1..1000
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).

Cf. A000041, A000700, A001560, A052001, A052003.
The strict version is A001318, with complement A090864.
The version for prime instead of odd numbers is A046063.
The version for squarefree instead of odd numbers is A038630.
The version for set partitions appears to be A032766.
The version for factorizations is A331050.
The version for strict factorizations is A331230.

import Data.List (findIndices)
a052002 n = a052002_list !! (n-1)
a052002_list = findIndices odd a000041_list
-- Reinhard Zumkeller, Nov 03 2015

N:= 1000: # to get all terms <= N
V:= Vector(N+1):
V[1]:= 1:
for i from 1 to (N+1)/2  do
  V[2*i..N+1]:= V[2*i..N+1] + V[1..N-2*i+2] mod 2
od:
select(t -> V[t+1]=1, [$1..N]); # Robert Israel, Jan 22 2017

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[2, k], {k, 0, 1}] (* Clark Kimberling, Jan 05 2014 *)

for(n=0, 200, if(numbpart(n)%2==1, print1(n", "))) \\ Altug Alkan, Nov 02 2015

From Peter Bala, Jan 22 2016: (Start)

Sum_{n>=0} x^a(n) = (1 + x)*(1 + x^3)*(1 + x^5)*... taken modulo 2. Proof: Product_{n>=1} 1 + x^(2*n-1) = Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(2*n-1)) = Product_{n>=1} (1 - x^(2*n))*(1 - x^(4*n-2))/( (1 - x^(2*n)) * (1 - x^(2*n-1)) ) = ( 1 + 2*Sum_{n>=1} (-1)^n*x^(2*n^2) )/(Product_{n>=1} (1 - x^n)) == 1/( Product_{n>=1} (1 - x^n) ) (mod 2). (End)

Offset corrected and b-file adjusted by Reinhard Zumkeller, Nov 03 2015

A111036 Numbers n such that p(6n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

1, 6, 22, 28, 31, 36, 61, 83, 91, 181, 216, 263, 356, 417, 418, 428, 528, 557, 777, 1133, 1243, 1408, 2170, 2708, 3046, 3867, 5100, 5540, 5662, 7418, 9397, 12110, 12797, 14787, 16161, 16482, 18022, 19431, 19667, 21180, 22011, 22720, 23560, 27903

Offset: 1

Parthasarathy Nambi, Nov 11 2005

nonn

			If n=91 then p(6n) = 27833079238879849385687 (prime).

Max Alekseyev, Table of n, a(n) for n = 1..782

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

Select[ Range[33333], PrimeQ[ PartitionsP[6# ]] &] (* Robert G. Wilson v *)

is(n)=isprime(numbpart(6*n)) \\ Charles R Greathouse IV, Feb 17 2017

a(10)-a(45) from Robert G. Wilson v, Nov 14 2005

A111045 Numbers n such that P(4n) is prime, where P(m) is the number of partitions of m.

Original entry on oeis.org

1, 9, 33, 42, 47, 53, 54, 110, 324, 534, 627, 642, 683, 728, 792, 1114, 2112, 2228, 2323, 2770, 3007, 3255, 3368, 3760, 4062, 4569, 6139, 7650, 7939, 8138, 8310, 8493, 8674, 9122, 9407, 10345, 11127, 13343, 14713, 15442, 15632, 16358, 16904, 18165, 19303

Offset: 1

Parthasarathy Nambi, Nov 11 2005

nonn

			If n=110 then P(4*n) = 74878248419470886233 (prime).

Max Alekseyev, Table of n, a(n) for n = 1..1271

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

Select[ Range[19923], PrimeQ[ PartitionsP[4# ]] &] (* Robert G. Wilson v *)

is(n)=isprime(numbpart(4*n)) \\ Charles R Greathouse IV, Feb 17 2017

a(9)-a(37) from Robert G. Wilson v, Nov 14 2005

A111389 Numbers n such that p(3n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

1, 2, 12, 44, 56, 62, 72, 122, 139, 166, 175, 182, 245, 251, 275, 362, 432, 526, 712, 831, 834, 836, 856, 909, 957, 1009, 1056, 1114, 1554, 2266, 2486, 2816, 3967, 4340, 5416, 6092, 6837, 6959, 7215, 7255, 7439, 7734, 9655, 10200, 11080, 11324, 11361, 12819

Offset: 1

Parthasarathy Nambi, Nov 09 2005

nonn

			If n=72 then p(3n) = 15285151248481 (prime).

Max Alekseyev, Table of n, a(n) for n = 1..1669

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

Select[ Range[13370], PrimeQ[ PartitionsP[3# ]] &] (* Robert G. Wilson v *)

is(n)=isprime(numbpart(3*n)) \\ Charles R Greathouse IV, Feb 17 2017

Elements of A046063 which are == 0 (mod 3) divided by 3

a(8)-a(48) from Robert G. Wilson v, Nov 11 2005

A114165 Numbers n such that p(2n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

1, 2, 3, 18, 66, 84, 93, 94, 106, 108, 151, 183, 220, 249, 273, 329, 543, 648, 789, 793, 1068, 1251, 1254, 1284, 1366, 1456, 1549, 1584, 1671, 1771, 2059, 2131, 2228, 2331, 2501, 3399, 3729, 4224, 4456, 4646, 4999, 5093, 5540, 6014, 6510, 6736, 7520, 8124

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

2n-th partition number (A000041(2n)) is prime.

Max Alekseyev, Table of n, a(n) for n = 1..2451

Cf. A000041, A046063, A068413, A114165, A111389, A111045, A114166, A111036, A114167, A114168, A114169, A114170.

Select[ Range[9137], PrimeQ[ PartitionsP[2# ]] &]

is(n)=isprime(numbpart(2*n)) \\ Charles R Greathouse IV, Feb 17 2017

A114166 Numbers n such that p(5n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

1, 88, 105, 147, 165, 217, 1481, 2216, 2579, 2604, 3008, 3658, 3694, 4329, 4353, 4447, 4534, 5074, 5793, 6120, 6578, 6648, 7861, 7994, 8276, 8851, 9421, 10371, 12350, 12359, 12389, 13010, 13345, 13479, 14532, 14727, 16461, 19313, 19466, 20354

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1252

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

Select[ Range[20780], PrimeQ[PartitionsP[5# ]] &]

is(n)=isprime(numbpart(5*n)) \\ Charles R Greathouse IV, Feb 17 2017

A114167 Numbers n such that p(7n) is prime, where p(n) is the number of partitions of n.

Original entry on oeis.org

11, 24, 75, 78, 94, 105, 155, 211, 293, 416, 506, 666, 1860, 3013, 3508, 3811, 4869, 5615, 5710, 8824, 8841, 8998, 10380, 11014, 11779, 13795, 14276, 15285, 18014, 19456, 19855, 22435, 23343, 23391, 26328, 30608, 31380, 32074, 32810, 33459

Offset: 1

Robert G. Wilson v, Nov 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..810

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

Select[ Range[28571], PrimeQ[PartitionsP[7# ]] &]

is(n)=isprime(numbpart(7*n)) \\ Charles R Greathouse IV, Feb 17 2017

cp's OEIS Frontend

A049575 Prime partition numbers.

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A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

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

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A052002 Numbers with an odd number of partitions.

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A111036 Numbers n such that p(6n) is prime, where p(n) is the number of partitions of n.

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A111045 Numbers n such that P(4n) is prime, where P(m) is the number of partitions of m.

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A111389 Numbers n such that p(3n) is prime, where p(n) is the number of partitions of n.

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