cp's OEIS Frontend

A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.

%I A372431 #5 May 05 2024 08:55:37
%S A372431 1,2,4,7,8,9,10,11,12,13,16,17,19,21,23,24,25,26,29,31,32,33,34,35,36,
%T A372431 37,38,40,41,43,44,46,47,48,49,50,53,57,58,59,61,62,64,65,67,69,71,72,
%U A372431 73,74,76,79,80,81,82,83,84,86,89,92,93,94,96,97,98,101
%N A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.
%C A372431 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 A372431 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.
%e A372431 The binary indices of 65 are {1,7}, and the prime indices are {3,6}, so 65 is in the sequence.
%e A372431 The terms together with their prime indices begin:
%e A372431      1: {}
%e A372431      2: {1}
%e A372431      4: {1,1}
%e A372431      7: {4}
%e A372431      8: {1,1,1}
%e A372431      9: {2,2}
%e A372431     10: {1,3}
%e A372431     11: {5}
%e A372431     12: {1,1,2}
%e A372431     13: {6}
%e A372431     16: {1,1,1,1}
%e A372431 The terms together with their binary expansions and binary indices begin:
%e A372431    1:       1 ~ {1}
%e A372431    2:      10 ~ {2}
%e A372431    4:     100 ~ {3}
%e A372431    7:     111 ~ {1,2,3}
%e A372431    8:    1000 ~ {4}
%e A372431    9:    1001 ~ {1,4}
%e A372431   10:    1010 ~ {2,4}
%e A372431   11:    1011 ~ {1,2,4}
%e A372431   12:    1100 ~ {3,4}
%e A372431   13:    1101 ~ {1,3,4}
%e A372431   16:   10000 ~ {5}
%t A372431 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A372431 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A372431 Select[Range[100],Intersection[bix[#],prix[#]]=={}&]
%Y A372431 For subset instead of disjoint we have A372430.
%Y A372431 The complement is A372432.
%Y A372431 Equal lengths: A071814, zeros of A372441.
%Y A372431 Equal sums: A372427, zeros of A372428.
%Y A372431 Equal maxima: A372436, zeros of A372442.
%Y A372431 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372431 A029837 gives greatest binary index, least A001511.
%Y A372431 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A372431 A061395 gives greatest prime index, least A055396.
%Y A372431 A070939 gives length of binary expansion.
%Y A372431 A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y A372431 Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.
%K A372431 nonn,base
%O A372431 1,2
%A A372431 _Gus Wiseman_, May 03 2024