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A325792
Positive integers with as many proper divisors as the sum of their prime indices.
18
1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 204, 234, 256, 260, 294, 308, 315, 350, 392, 416, 486, 500, 512, 690, 696, 798, 920, 1024, 1026, 1064, 1088, 1116, 1122, 1190, 1365, 1430, 1458, 1496, 1755, 1936, 1968, 2025, 2048, 2058, 2079
OFFSET
1,2
COMMENTS
First differs from A325780 in having 204.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).
EXAMPLE
The term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.
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}
204: {1,1,2,7}
234: {1,2,2,6}
256: {1,1,1,1,1,1,1,1}
MATHEMATICA
Select[Range[100], DivisorSigma[0, #]-1==Total[Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]*k]]&]
CROSSREFS
Positions of 1's in A325794.
Heinz numbers of the partitions counted by A325828.
Sequence in context: A077569 A270140 A333020 * A325780 A283423 A073935
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 23 2019
STATUS
approved