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allocated for Gus WisemanNumber of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.
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, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156
0,8
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.
The a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
3311 711 61111 521111 5511 B11 A1111
321111 3221111 9111 721111 731111
531111 811111 33221111
3321111 5221111 422111111
22221111 43111111
42111111
red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n], Times@@red/@#==Times@@Length/@Split[#]&]], {n, 0, 30}]
The LHS (product of multiplicities) is A005361, counted by A266477.
The RHS (product of prime shadows) is A353394, first appearances A353397.
A related comparison is A353396, ranked by A353395.
These partitions are ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702, recursive version A353393.
A339095 counts partitions by product.
Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353426.
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Gus Wiseman, May 17 2022
approved
editing
allocated for Gus Wiseman
allocated
approved