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Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.
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%I #10 May 22 2024 17:00:58

%S 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 -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 -5,0,-1,2,-1,1,0,3,-3,2,1,1,0,2,1,4,-3,-1

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

%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.

%H Robert Israel, <a href="/A372441/b372441.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000120(n) - A001222(n).

%p f:= proc(n) convert(convert(n,base,2),`+`)-numtheory:-bigomega(n) end proc:

%p map(f, [$1..100]); # _Robert Israel_, May 22 2024

%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 Table[Length[bix[n]]-Length[prix[n]],{n,100}]

%Y Positions of zeros are A071814.

%Y For sum instead of length we have A372428, zeros A372427.

%Y For minimum instead of length we have A372437, zeros {}.

%Y For maximum instead of length we have A372442, zeros A372436.

%Y Positions of odd terms are A372590, even A372591.

%Y A003963 gives product of prime indices.

%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, A014499, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.

%K sign,base

%O 1,7

%A _Gus Wiseman_, May 07 2024