A051005 -id:A051005 - cp's OEIS frontend

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

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

Offset: 1

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

A051044 Odd values of the PartitionsQ function A000009.

Original entry on oeis.org

1, 1, 1, 3, 5, 15, 27, 89, 165, 585, 1113, 4097, 7917, 29927, 58499, 225585, 444793, 1741521, 3457027, 13699699, 27342421, 109420549, 219358315, 884987529, 1780751883, 7233519619, 14600965705, 59656252987, 120742510607, 495811828759, 1005862035461

Offset: 0

Eric W. Weisstein

nonn

A000009(n) is odd iff n is of the form k*(3*k - 1)/2 or k*(3*k + 1)/2. - Jonathan Vos Post, Jun 18 2005
Eric W. Weisstein comments: "The values of n for which A000009(n) is prime are 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, ... (A035359). These values correspond to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (A051005). It is not known if a(n) is infinitely often prime, but Gordon and Ono (1997) proved that it is 'almost always' divisible by any given power of 2 (1997)."
Semiprime values begin: a(5) = 15 = 3 * 5, a(11) = 4097 = 17 * 241, a(20) = 27342421 = 389 * 70289, a(24) = 1780751883 = 3 * 593583961, a(28) = 120742510607 = 31 * 3894919697. - Jonathan Vos Post, Jun 18 2005

Alois P. Heinz, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Partition Function Q Congruences

Cf. A000009, A001318, A035359, A051005, A118303.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> b((m->m*(3*m-1)/2)(ceil(-n*(-1)^n/2))):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2021

PartitionsQ /@ Table[n*((n + 1)/6), {n, Select[Range[50], Mod[#, 3] != 1 & ]}] (* Jean-François Alcover, Oct 31 2012, after Reinhard Zumkeller *)

a(n) = A000009(A001318(n)). - Reinhard Zumkeller, Apr 22 2006

Missing initial 1 inserted by Sean A. Irvine, Aug 23 2021

A368297 Prime plane partition numbers.

Original entry on oeis.org

3, 13, 859, 5668963, 12733429, 281846923, 10499640707, 776633557947931, 59206066030052023, 13621664240071959464038764694637, 27217095019687611064080107410267607999874139, 208912772327685894433117242327777497768893400876928857463950152067659

Offset: 1

Ilya Gutkovskiy, Dec 20 2023

nonn

Prime values of A000219.

Eric Weisstein's World of Mathematics, Plane Partition

Cf. A000219, A049575, A051005, A285216 (indices).

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
select(isprime, [seq(a(n), n=0..800)])[];  # Alois P. Heinz, Dec 20 2023

nmax = 750; Select[CoefficientList[Series[Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], PrimeQ]

a(n) = A000219(A285216(n)).

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

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A051044 Odd values of the PartitionsQ function A000009.

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A368297 Prime plane partition numbers.

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