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A325780
Heinz numbers of perfect integer partitions.
26
1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878
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 all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.
FORMULA
Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
32: {1,1,1,1,1}
42: {1,2,4}
54: {1,2,2,2}
56: {1,1,1,4}
64: {1,1,1,1,1,1}
100: {1,1,3,3}
128: {1,1,1,1,1,1,1}
162: {1,2,2,2,2}
176: {1,1,1,1,5}
234: {1,2,2,6}
256: {1,1,1,1,1,1,1,1}
260: {1,1,3,6}
MATHEMATICA
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0, #]-1]&]
CROSSREFS
Equals the sorted concatenation of the triangle A258119.
A subsequence of A299702 and A325781.
Sequence in context: A270140 A333020 A325792 * A283423 A073935 A325764
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 21 2019
STATUS
approved