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Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.
5

%I #6 May 05 2024 08:55:26

%S 1,3,5,15,27,39,55,63,85,121,125,135,169,171,175,209,243,247,255,299,

%T 375,399,437,459,507,539,605,637,725,735,783,841,867,891,1085,1215,

%U 1323,1331,1375,1519,1767,1815,1863,2079,2125,2187,2223,2295,2299,2331,2405

%N Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

%C 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.

%C 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.

%C 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.

%F Row k of A304038 is a subset of row k of A048793.

%e 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.

%e The terms together with their prime indices begin:

%e 1: {}

%e 3: {2}

%e 5: {3}

%e 15: {2,3}

%e 27: {2,2,2}

%e 39: {2,6}

%e 55: {3,5}

%e 63: {2,2,4}

%e 85: {3,7}

%e 121: {5,5}

%e 125: {3,3,3}

%e The terms together with their binary expansions and binary indices begin:

%e 1: 1 ~ {1}

%e 3: 11 ~ {1,2}

%e 5: 101 ~ {1,3}

%e 15: 1111 ~ {1,2,3,4}

%e 27: 11011 ~ {1,2,4,5}

%e 39: 100111 ~ {1,2,3,6}

%e 55: 110111 ~ {1,2,3,5,6}

%e 63: 111111 ~ {1,2,3,4,5,6}

%e 85: 1010101 ~ {1,3,5,7}

%e 121: 1111001 ~ {1,4,5,6,7}

%e 125: 1111101 ~ {1,3,4,5,6,7}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[1000],SubsetQ[bix[#],prix[#]]&]

%Y The version for equal lengths is A071814, zeros of A372441.

%Y The version for equal sums is A372427, zeros of A372428.

%Y For disjoint instead of subset we have A372431, complement A372432.

%Y The version for equal maxima is A372436, zeros of A372442.

%Y A019565 gives Heinz number of binary indices, adjoint A048675.

%Y A029837 gives greatest binary index, least A001511.

%Y A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

%Y A061395 gives greatest prime index, least A055396.

%Y A070939 gives length of binary expansion.

%Y A112798 lists prime indices, length A001222, reverse A296150, sum A056239.

%Y Cf. A000720, A001221, A230877, A243055, A304818, A355536, A358136, A372429.

%K nonn,base

%O 1,2

%A _Gus Wiseman_, May 02 2024