STATUS
proposed
approved
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Revision History for A331380
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Numbers whose sum of prime factors is divisible by their sum of prime indices.
(history; published version)
(history; published version)
STATUS
editing
proposed
CROSSREFS
Partitions whose Heinz number is divisible by their sum of primes are : A330953.
NAME
allocated for Gus WisemanNumbers whose sum of prime factors is divisible by their sum of prime indices.
DATA
2, 4, 8, 16, 32, 33, 39, 55, 64, 65, 66, 74, 77, 78, 86, 91, 110, 128, 130, 132, 154, 156, 164, 182, 188, 220, 256, 260, 264, 308, 312, 364, 371, 411, 440, 459, 512, 513, 520, 528, 530, 616, 624, 636, 689, 728, 746, 755, 765, 766, 855, 880, 906, 915, 918, 1007
OFFSET
1,1
COMMENTS
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.
EXAMPLE
The sequence of terms together with their prime indices begins:
2: {1}
4: {1,1}
8: {1,1,1}
16: {1,1,1,1}
32: {1,1,1,1,1}
33: {2,5}
39: {2,6}
55: {3,5}
64: {1,1,1,1,1,1}
65: {3,6}
66: {1,2,5}
74: {1,12}
77: {4,5}
78: {1,2,6}
86: {1,14}
91: {4,6}
110: {1,3,5}
128: {1,1,1,1,1,1,1}
130: {1,3,6}
132: {1,1,2,5}
For example, 132 has prime factors {2,2,3,11} and prime indices {1,1,2,5}, and 18 is divisible by 9, so 132 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], Divisible[Plus@@Prime/@primeMS[#], Plus@@primeMS[#]]&]
CROSSREFS
These are the Heinz numbers of the partitions counted by A331380.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes are A330953.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A330954, A331378, A331379, A331382.
KEYWORD
allocated
nonn
AUTHOR
Gus Wiseman, Jan 16 2020
STATUS
approved
editing
NAME
allocated for Gus Wiseman
KEYWORD
allocated
STATUS
approved