OFFSET
0,8
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.
EXAMPLE
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
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2022
STATUS
approved