cp's OEIS Frontend

A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

%I A372441 #10 May 22 2024 17:00:58
%S A372441 1,0,1,-1,1,0,2,-2,0,0,2,-1,2,1,2,-3,1,-1,2,-1,1,1,3,-2,1,1,1,0,3,1,4,
%T A372441 -4,0,0,1,-2,2,1,2,-2,2,0,3,0,1,2,4,-3,1,0,2,0,3,0,3,-1,2,2,4,0,4,3,3,
%U A372441 -5,0,-1,2,-1,1,0,3,-3,2,1,1,0,2,1,4,-3,-1
%N A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.
%C A372441 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 A372441 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.
%H A372441 Robert Israel, <a href="/A372441/b372441.txt">Table of n, a(n) for n = 1..10000</a>
%F A372441 a(n) = A000120(n) - A001222(n).
%p A372441 f:= proc(n) convert(convert(n,base,2),`+`)-numtheory:-bigomega(n) end proc:
%p A372441 map(f, [$1..100]); # _Robert Israel_, May 22 2024
%t A372441 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A372441 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A372441 Table[Length[bix[n]]-Length[prix[n]],{n,100}]
%Y A372441 Positions of zeros are A071814.
%Y A372441 For sum instead of length we have A372428, zeros A372427.
%Y A372441 For minimum instead of length we have A372437, zeros {}.
%Y A372441 For maximum instead of length we have A372442, zeros A372436.
%Y A372441 Positions of odd terms are A372590, even A372591.
%Y A372441 A003963 gives product of prime indices.
%Y A372441 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372441 A029837 gives greatest binary index, least A001511.
%Y A372441 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A372441 A061395 gives greatest prime index, least A055396.
%Y A372441 A070939 gives length of binary expansion.
%Y A372441 A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y A372441 Cf. A000720, A014499, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.
%K A372441 sign,base
%O A372441 1,7
%A A372441 _Gus Wiseman_, May 07 2024