cp's OEIS Frontend

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

%I A353398 #5 May 20 2022 08:51:14
%S A353398 1,1,0,0,1,1,1,2,1,2,1,2,6,5,4,4,6,6,8,8,13,16,13,16,18,16,20,21,27,
%T A353398 30,27,33,41,44,51,48,58,61,66,66,74,83,86,99,102,111,115,126,137,147,
%U A353398 156
%N A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.
%C A353398 We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
%e A353398 The a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
%e A353398   3311  711     61111  521111   5511      B11       A1111
%e A353398         321111         3221111  9111      721111    731111
%e A353398                                 531111    811111    33221111
%e A353398                                 3321111   5221111   422111111
%e A353398                                 22221111  43111111
%e A353398                                 42111111
%t A353398 red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t A353398 Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]
%Y A353398 The LHS (product of multiplicities) is A005361, counted by A266477.
%Y A353398 The RHS (product of prime shadows) is A353394, first appearances A353397.
%Y A353398 A related comparison is A353396, ranked by A353395.
%Y A353398 These partitions are ranked by A353399.
%Y A353398 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353398 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353398 A124010 gives prime signature, sorted A118914.
%Y A353398 A181819 gives prime shadow, with an inverse A181821.
%Y A353398 A325131 lists numbers relatively prime to their prime shadow.
%Y A353398 A325755 lists numbers divisible by their prime shadow, counted by A325702.
%Y A353398 A339095 counts partitions by product (or factorizations by sum).
%Y A353398 Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.
%K A353398 nonn
%O A353398 0,8
%A A353398 _Gus Wiseman_, May 17 2022