cp's OEIS Frontend

A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.

%I A331383 #12 Feb 14 2025 05:21:30
%S A331383 0,0,0,0,0,0,1,0,2,2,1,1,1,2,2,2,1,4,2,2,2,4,2,3,4,1,3,4,5,0,3,3,1,6,
%T A331383 2,1,5,4,2,3,4,2,2,3,1,5,2,3,4,6,5,2,7,1,3,5,3,4,2,5,5,4,7,3,6,4,4,2,
%U A331383 4,4,3,9,4,3,5,3,5,4,4,4,3,7,4,2,8,2,3
%N A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.
%e A331383 The a(n) partitions for n = 7, 9, 18, 24:
%e A331383   (4,3)  (6,3)    (12,4,1,1)                 (19,4,1)
%e A331383          (4,4,1)  (11,4,1,1,1)               (18,4,1,1)
%e A331383                   (8,5,1,1,1,1,1)            (9,6,1,1,1,1,1,1,1,1,1)
%e A331383                   (4,2,2,2,1,1,1,1,1,1,1,1)
%e A331383 For example, (4,4,1) has sum of primes of parts 7+7+2 = 16 and product of parts 4*4*1 = 16, so is counted under a(9).
%t A331383 Table[Length[Select[IntegerPartitions[n],Times@@#==Plus@@Prime/@#&]],{n,30}]
%o A331383 (PARI) a(n) = my(c=0); forpart(v=n, if(vecprod(Vec(v))==sum(i=1, #v, prime(v[i])), c++)); c; \\ _Jinyuan Wang_, Feb 14 2025
%Y A331383 The Heinz numbers of these partitions are given by A331384.
%Y A331383 Numbers divisible by the sum of their prime factors are A036844.
%Y A331383 Partitions whose product is divisible by their sum are A057568.
%Y A331383 Numbers divisible by the sum of their prime indices are A324851.
%Y A331383 Product of prime indices is divisible by sum of prime indices: A326149.
%Y A331383 Partitions whose Heinz number is divisible by their sum are A330950.
%Y A331383 Partitions whose Heinz number is divisible by their sum of primes: A330953.
%Y A331383 Sum of prime factors is divisible by sum of prime indices: A331380
%Y A331383 Partitions whose product divides their sum of primes are A331381.
%Y A331383 Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.
%K A331383 nonn
%O A331383 1,9
%A A331383 _Gus Wiseman_, Jan 16 2020
%E A331383 a(71)-a(87) from _Robert Price_, Apr 10 2020