A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

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, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964

Offset: 0

Author

Gus Wiseman, May 15 2022

Keywords

• nonn

Comments

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.

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) = 9 partitions (A..D = 10..13):
  (53)  (72)    (73)    (B)     (75)     (D)      (B3)
        (621)   (532)   (A1)    (651)    (B2)     (752)
        (4221)  (631)   (4331)  (732)    (A21)    (761)
                (4411)          (6321)   (43321)  (A31)
                                (6411)   (44311)  (C11)
                                (43221)           (6521)
                                (44211)           (9221)
                                                  (54221)
                                                  (64211)
		

Crossrefs

The LHS (prime shadow) is A181819, with an inverse A181821.

The RHS (product of prime shadows) is A353394, first appearances A353397.

These partitions are ranked by A353395.

A related comparison is A353398, ranked by A353399.

A001222 counts prime factors with multiplicity, distinct A001221.

A003963 gives product of prime indices.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914, product A005361.

A239455 counts Look-and-Say partitions, ranked by A351294.

A324850 lists numbers divisible by the product of their prime indices.

Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.

Programs

Mathematica

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]