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A325781
Heinz numbers of complete integer partitions.
55
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 160, 162, 168, 176, 180, 192, 198, 200, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288, 294, 300
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[#]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 21 2019
STATUS
approved