A284114 Number of partitions of n such that Omega(n) (= number of prime divisors of n counted with multiplicity) equals the sum of Omega of all parts; Omega = A001222.

1, 1, 1, 2, 2, 3, 5, 4, 7, 9, 12, 5, 22, 6, 17, 19, 55, 7, 50, 8, 60, 28, 32, 9, 166, 37, 41, 113, 122, 10, 137, 11, 631, 51, 56, 57, 475, 12, 64, 66, 620, 13, 258, 14, 282, 301, 83, 15, 2229, 90, 359, 95, 394, 16, 1302, 105, 1435, 109, 114, 17, 1708, 18, 125

Offset: 0

Author

Alois P. Heinz, Mar 20 2017

Keywords

• nonn

Examples

			a(5) = 3: [2,1,1,1], [3,1,1], [5].
a(6) = 5: [2,2,1,1], [3,2,1], [3,3], [4,1,1], [6].
a(7) = 4: [2,1,1,1,1,1], [3,1,1,1,1], [5,1,1], [7].
a(8) = 7: [2,2,2,1,1], [3,2,2,1], [3,3,2], [4,2,1,1], [4,3,1], [6,2], [8].
a(9) = 9: [2,2,1,1,1,1,1], [3,2,1,1,1,1], [3,3,1,1,1], [4,1,1,1,1,1], [5,2,1,1], [5,3,1], [6,1,1,1], [7,2], [9].
a(10) = 12: [2,2,1,1,1,1,1,1], [3,2,1,1,1,1,1], [3,3,1,1,1,1], [4,1,1,1,1,1,1], [5,2,1,1,1], [5,3,1,1], [5,5], [6,1,1,1,1], [7,2,1], [7,3], [9,1], [10].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..6000
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A000040, A000041, A000720, A001222, A283528.

Programs

Maple

with(numtheory):
b:= proc(n, i, m) option remember; `if`(n=0 or i=1,
      `if`(m=0, 1, 0), `if`(m<0, 0, b(n, i-1, m)+
      `if`(i>n, 0, b(n-i, i, m-bigomega(i)))))
    end:
a:= n-> b(n$2, bigomega(n)):
seq(a(n), n=0..80);

Mathematica

b[n_, i_, m_] := b[n, i, m] = If[n == 0 || i == 1, If[m == 0, 1, 0], If[m < 0, 0, b[n, i-1, m] + If[i>n, 0, b[n-i, i, m - PrimeOmega[i]]]]];
a[0] = 1; a[n_] := b[n, n, PrimeOmega[n]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Mar 25 2017, translated from Maple *)

PARI

b(n, i, m) = if(n==0 || i==1, if(m==0, 1, 0), if(m<0, 0, b(n, i - 1, m) + if(i>n, 0, b(n - i, i, m - bigomega(i)))));
a(n) = if(n<1, 1, b(n, n, bigomega(n)));
for(n=0, 80, print1(a(n),", ")) \\ Indranil Ghosh, Mar 25 2017, translated from Mathematica

Formula

a(p) = A000720(p) for prime p.