A353398 Number 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

Offset: 0

Author

Gus Wiseman, May 17 2022

Keywords

• nonn

Comments

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.

Examples

			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
		

Crossrefs

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.

A339095 counts partitions by product (or factorizations by sum).

Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.

Programs

Mathematica

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]