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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.

%I A284114 #17 Mar 25 2017 20:36:03
%S A284114 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,
%T A284114 113,122,10,137,11,631,51,56,57,475,12,64,66,620,13,258,14,282,301,83,
%U A284114 15,2229,90,359,95,394,16,1302,105,1435,109,114,17,1708,18,125
%N 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.
%H A284114 Alois P. Heinz, <a href="/A284114/b284114.txt">Table of n, a(n) for n = 0..6000</a>
%H A284114 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A284114 a(p) = A000720(p) for prime p.
%e A284114 a(5) = 3: [2,1,1,1], [3,1,1], [5].
%e A284114 a(6) = 5: [2,2,1,1], [3,2,1], [3,3], [4,1,1], [6].
%e A284114 a(7) = 4: [2,1,1,1,1,1], [3,1,1,1,1], [5,1,1], [7].
%e A284114 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].
%e A284114 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].
%e A284114 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].
%p A284114 with(numtheory):
%p A284114 b:= proc(n, i, m) option remember; `if`(n=0 or i=1,
%p A284114       `if`(m=0, 1, 0), `if`(m<0, 0, b(n, i-1, m)+
%p A284114       `if`(i>n, 0, b(n-i, i, m-bigomega(i)))))
%p A284114     end:
%p A284114 a:= n-> b(n$2, bigomega(n)):
%p A284114 seq(a(n), n=0..80);
%t A284114 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]]]]];
%t A284114 a[0] = 1; a[n_] := b[n, n, PrimeOmega[n]];
%t A284114 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Mar 25 2017, translated from Maple *)
%o A284114 (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)))));
%o A284114 a(n) = if(n<1, 1, b(n, n, bigomega(n)));
%o A284114 for(n=0, 80, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 25 2017, translated from Mathematica
%Y A284114 Cf. A000040, A000041, A000720, A001222, A283528.
%K A284114 nonn
%O A284114 0,4
%A A284114 _Alois P. Heinz_, Mar 20 2017