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A366321
Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.
4
1, 3, 7, 10, 13, 16, 19, 21, 22, 27, 28, 29, 34, 36, 37, 39, 43, 46, 48, 52, 53, 55, 57, 61, 62, 64, 66, 71, 75, 76, 79, 81, 82, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 113, 115, 116, 117, 118, 120, 129, 130, 131, 133, 134, 136, 138, 139, 144
OFFSET
0,2
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 prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.
The terms together with their prime indices begin:
1: {}
3: {2}
7: {4}
10: {1,3}
13: {6}
16: {1,1,1,1}
19: {8}
21: {2,4}
22: {1,5}
27: {2,2,2}
28: {1,1,4}
29: {10}
34: {1,7}
36: {1,1,2,2}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], EvenQ[Total[prix[#]]]&&FreeQ[prix[#], Total[prix[#]]/2]&]
CROSSREFS
Partitions of this type are counted by A182616, strict A365828.
A066207 lists numbers with all even prime indices, odd A066208.
A086543 lists numbers with at least one odd prime index, counted by A366322.
A300063 ranks partitions of odd numbers.
A366319 ranks partitions of n not containing n/2.
A366321 ranks partitions of 2k that do not contain k.
Sequence in context: A103541 A183178 A340604 * A310183 A145004 A310184
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 13 2023
STATUS
approved