A382290 a(n) = A064547(n) - A001221(n).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset

1

Comments

First differs from A295662 at n = 64, and from A367512 at n = 128.

Analogous to prime excess (A046660): the excess of the number of Fermi-Dirac factors of n over the number of distinct prime factors of n.

The first positions of a(n) = 0, 1, 2, 3, ..., are n = 1, 8, 128, 3456, 279936, 34992000, 8957952000, ... (A382293).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Additive with a(p^e) = A000120(e) - 1 = A048881(e-1).

a(n) = log_2(A382291(n)) = log_2(A037445(n)/A034444(n)).

a(n) >= 0, with equality if and only if n is in A138302.

a(n) = 1 if and only if n is in A382292.

Sum_{k=1..n} a(k) ~ c * n, c = 0.1360544... (A382294).

Mathematica

f[p_, e_] := DigitCount[e, 2, 1] - 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecsum(apply(x -> hammingweight(x) - 1, factor(n)[, 2]));

Crossrefs

Cf. A000120, A001221, A034444, A037445, A046660, A048881, A064547, A138302, A382291, A382292, A382293, A382294.

Cf. A295662, A367512.

Sequence in context: A366124 A295662 A367512 * A382424 A187946 A330549

Adjacent sequences: A382287 A382288 A382289 * A382291 A382292 A382293

Keyword

nonn,easy,base

Author

Amiram Eldar, Mar 21 2025

Status

approved