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

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

Offset

1,7

Comments

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.

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

Maple

f:= proc(n) convert(convert(n, base, 2), `+`)-numtheory:-bigomega(n) end proc:
map(f, [$1..100]); # Robert Israel, May 22 2024

Mathematica

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[bix[n]]-Length[prix[n]], {n, 100}]

Crossrefs

Positions of zeros are A071814.

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

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

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

Positions of odd terms are A372590, even A372591.

A003963 gives product of prime indices.

A019565 gives Heinz number of binary indices, adjoint A048675.

A029837 gives greatest binary index, least A001511.

A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

A061395 gives greatest prime index, least A055396.

A070939 gives length of binary expansion.

A112798 lists prime indices, length A001222, reverse A296150, sum A056239.

Cf. A000720, A014499, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.

Sequence in context: A113406 A151851 A321447 * A341698 A357069 A033461

Adjacent sequences: A372438 A372439 A372440 * A372442 A372443 A372444

Keyword

sign,base

Author

Gus Wiseman, May 07 2024

Status

approved