OFFSET
0,7
EXAMPLE
The a(0) = 1 through a(18) = 2 partitions:
n= 0: ()
n= 1: (1)
n= 2:
n= 3:
n= 4: (211)
n= 5:
n= 6: (3111) (2211)
n= 7:
n= 8: (41111)
n= 9:
n=10: (511111)
n=11: (32111111)
n=12: (6111111) (22221111)
n=13: (322111111)
n=14: (71111111) (4211111111)
n=15:
n=16: (811111111) (4411111111) (42211111111)
n=17: (521111111111) (332111111111) (322211111111)
n=18: (9111111111) (333111111111)
For example, the partition y = (322111111) has multiplicities (1,2,6) with product 12, and the product of parts is also 3*2*2*1*1*1*1*1*1 = 12, so y is counted under a(13).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@Length/@Split[#]&]], {n, 0, 30}]
PROG
(PARI) a(n) = {my(nb=0); forpart(p=n, my(s=Set(p), v=Vec(p)); if (vecprod(vector(#s, i, #select(x->(x==s[i]), v))) == vecprod(v), nb++); ); nb; } \\ Michel Marcus, May 20 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2022
EXTENSIONS
a(71)-a(100) from Alois P. Heinz, May 20 2022
STATUS
approved