OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The sum of prime indices of n is A056239(n). A number is in this sequence iff its divisors have sums of prime indices covering an initial interval of nonnegative integers. For example, the divisors of 60 are {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, with respective sums of prime indices {0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7}, so 60 is in the sequence.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
30: {1,2,3}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
48: {1,1,1,1,2}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
MATHEMATICA
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], normQ[hwt/@Rest[Divisors[#]]]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 21 2019
STATUS
approved