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A372432
Positive integers k such that the prime indices of k are not disjoint from the binary indices of k.
8
3, 5, 6, 14, 15, 18, 20, 22, 27, 28, 30, 39, 42, 45, 51, 52, 54, 55, 56, 60, 63, 66, 68, 70, 75, 77, 78, 85, 87, 88, 90, 91, 95, 99, 100, 102, 104, 105, 110, 111, 114, 117, 119, 121, 123, 125, 126, 133, 135, 138, 140, 147, 150, 152, 154, 159, 162, 165, 168
OFFSET
1,1
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 binary indices of 18 are {2,5}, and the prime indices are {1,2,2}, so 18 is in the sequence.
The terms together with their prime indices begin:
3: {2}
5: {3}
6: {1,2}
14: {1,4}
15: {2,3}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
27: {2,2,2}
28: {1,1,4}
30: {1,2,3}
The terms together with their binary expansions and binary indices begin:
3: 11 ~ {1,2}
5: 101 ~ {1,3}
6: 110 ~ {2,3}
14: 1110 ~ {2,3,4}
15: 1111 ~ {1,2,3,4}
18: 10010 ~ {2,5}
20: 10100 ~ {3,5}
22: 10110 ~ {2,3,5}
27: 11011 ~ {1,2,4,5}
28: 11100 ~ {3,4,5}
30: 11110 ~ {2,3,4,5}
MATHEMATICA
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Intersection[bix[#], prix[#]]!={}&]
CROSSREFS
For subset instead of overlap we have A372430.
The complement is A372431.
Equal lengths: A071814, zeros of A372441.
Equal sums: A372427, zeros of A372428.
Equal maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Sequence in context: A048968 A355956 A185737 * A092862 A112863 A333292
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, May 03 2024
STATUS
approved