cp's OEIS Frontend

A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

%I A372430 #6 May 05 2024 08:55:26
%S A372430 1,3,5,15,27,39,55,63,85,121,125,135,169,171,175,209,243,247,255,299,
%T A372430 375,399,437,459,507,539,605,637,725,735,783,841,867,891,1085,1215,
%U A372430 1323,1331,1375,1519,1767,1815,1863,2079,2125,2187,2223,2295,2299,2331,2405
%N A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.
%C A372430 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 A372430 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 A372430 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 A372430 Row k of A304038 is a subset of row k of A048793.
%e A372430 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 A372430 The terms together with their prime indices begin:
%e A372430      1: {}
%e A372430      3: {2}
%e A372430      5: {3}
%e A372430     15: {2,3}
%e A372430     27: {2,2,2}
%e A372430     39: {2,6}
%e A372430     55: {3,5}
%e A372430     63: {2,2,4}
%e A372430     85: {3,7}
%e A372430    121: {5,5}
%e A372430    125: {3,3,3}
%e A372430 The terms together with their binary expansions and binary indices begin:
%e A372430      1:              1 ~ {1}
%e A372430      3:             11 ~ {1,2}
%e A372430      5:            101 ~ {1,3}
%e A372430     15:           1111 ~ {1,2,3,4}
%e A372430     27:          11011 ~ {1,2,4,5}
%e A372430     39:         100111 ~ {1,2,3,6}
%e A372430     55:         110111 ~ {1,2,3,5,6}
%e A372430     63:         111111 ~ {1,2,3,4,5,6}
%e A372430     85:        1010101 ~ {1,3,5,7}
%e A372430    121:        1111001 ~ {1,4,5,6,7}
%e A372430    125:        1111101 ~ {1,3,4,5,6,7}
%t A372430 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A372430 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A372430 Select[Range[1000],SubsetQ[bix[#],prix[#]]&]
%Y A372430 The version for equal lengths is A071814, zeros of A372441.
%Y A372430 The version for equal sums is A372427, zeros of A372428.
%Y A372430 For disjoint instead of subset we have A372431, complement A372432.
%Y A372430 The version for equal maxima is A372436, zeros of A372442.
%Y A372430 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372430 A029837 gives greatest binary index, least A001511.
%Y A372430 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A372430 A061395 gives greatest prime index, least A055396.
%Y A372430 A070939 gives length of binary expansion.
%Y A372430 A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y A372430 Cf. A000720, A001221, A230877, A243055, A304818, A355536, A358136, A372429.
%K A372430 nonn,base
%O A372430 1,2
%A A372430 _Gus Wiseman_, May 02 2024