A372431 - OEIS

%I #5 May 05 2024 08:55:37

%S 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 37,38,40,41,43,44,46,47,48,49,50,53,57,58,59,61,62,64,65,67,69,71,72,

%U 73,74,76,79,80,81,82,83,84,86,89,92,93,94,96,97,98,101

%N Positive integers k such that the prime indices of k are disjoint from 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.

%e The binary indices of 65 are {1,7}, and the prime indices are {3,6}, so 65 is in the sequence.

%e The terms together with their prime indices begin:

%e      1: {}

%e      2: {1}

%e      4: {1,1}

%e      7: {4}

%e      8: {1,1,1}

%e      9: {2,2}

%e     10: {1,3}

%e     11: {5}

%e     12: {1,1,2}

%e     13: {6}

%e     16: {1,1,1,1}

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

%e    1:       1 ~ {1}

%e    2:      10 ~ {2}

%e    4:     100 ~ {3}

%e    7:     111 ~ {1,2,3}

%e    8:    1000 ~ {4}

%e    9:    1001 ~ {1,4}

%e   10:    1010 ~ {2,4}

%e   11:    1011 ~ {1,2,4}

%e   12:    1100 ~ {3,4}

%e   13:    1101 ~ {1,3,4}

%e   16:   10000 ~ {5}

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

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

%t Select[Range[100],Intersection[bix[#],prix[#]]=={}&]

%Y For subset instead of disjoint we have A372430.

%Y The complement is A372432.

%Y Equal lengths: A071814, zeros of A372441.

%Y Equal sums: A372427, zeros of A372428.

%Y Equal maxima: 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, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

%K nonn,base

%O 1,2

%A _Gus Wiseman_, May 03 2024