OFFSET
1,2
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.
Conjecture: The only number whose binary indices are a subset of its prime indices is 4100, with binary indices {3,13} and prime indices {1,1,3,3,13}. Verified up to 10,000,000.
EXAMPLE
The prime indices of 135 are {2,2,2,3}, and the binary indices are {1,2,3,8}. Since {2,3} is a subset of {1,2,3,8}, 135 is in the sequence.
The terms together with their prime indices begin:
1: {}
3: {2}
5: {3}
15: {2,3}
27: {2,2,2}
39: {2,6}
55: {3,5}
63: {2,2,4}
85: {3,7}
121: {5,5}
125: {3,3,3}
The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
3: 11 ~ {1,2}
5: 101 ~ {1,3}
15: 1111 ~ {1,2,3,4}
27: 11011 ~ {1,2,4,5}
39: 100111 ~ {1,2,3,6}
55: 110111 ~ {1,2,3,5,6}
63: 111111 ~ {1,2,3,4,5,6}
85: 1010101 ~ {1,3,5,7}
121: 1111001 ~ {1,4,5,6,7}
125: 1111101 ~ {1,3,4,5,6,7}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[1000], SubsetQ[bix[#], prix[#]]&]
CROSSREFS
A070939 gives length of binary expansion.
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, May 02 2024
STATUS
approved