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%I #5 May 17 2022 17:51:08
%S 1,0,1,1,0,2,0,3,1,3,4,3,7,5,9,8,12,15,15,20,21,25,31,33,38,42,46,56,
%T 61,67,78,76,96,100,114,131,130,157,157,185,200,214,236,253,275,302,
%U 333,351,386,408,440,486,515,564,596,633,691,734,800,854,899,964
%N Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C 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 The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
%e (53) (72) (73) (B) (75) (D) (B3)
%e (621) (532) (A1) (651) (B2) (752)
%e (4221) (631) (4331) (732) (A21) (761)
%e (4411) (6321) (43321) (A31)
%e (6411) (44311) (C11)
%e (43221) (6521)
%e (44211) (9221)
%e (54221)
%e (64211)
%t red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]
%Y The LHS (prime shadow) is A181819, with an inverse A181821.
%Y The RHS (product of prime shadows) is A353394, first appearances A353397.
%Y These partitions are ranked by A353395.
%Y A related comparison is A353398, ranked by A353399.
%Y A001222 counts prime factors with multiplicity, distinct A001221.
%Y A003963 gives product of prime indices.
%Y A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A124010 gives prime signature, sorted A118914, product A005361.
%Y A239455 counts Look-and-Say partitions, ranked by A351294.
%Y A324850 lists numbers divisible by the product of their prime indices.
%Y Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.
%K nonn
%O 0,6
%A _Gus Wiseman_, May 15 2022