A366371 Number of divisors of n that are either an odd prime power > 1 or twice such a number.

0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 2, 1, 2, 2, 0, 1, 4, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 4, 1, 0, 2, 2, 2, 4, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 6, 2, 2, 2, 2, 1, 4, 1, 2, 3, 0, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 3, 2, 2, 4, 1, 2, 4, 2, 1, 4, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 4, 3, 4, 1, 4, 1, 2, 3

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = Sum_{d|n} (1-A354108(d)).

For odd n, a(n) = A001222(n), for even n, a(n) = 2*A001222(A000265(n)).

Prog

(PARI)
A354108(n) = ((n && !bitand(n, n-1)) || !isprimepower(n/(2-(n%2))));
A366371(n) = sumdiv(n, d, !A354108(d));
(Python)
from sympy import factorint
def A366371(n): return sum(factorint(n).values()) if n&1 else sum(factorint(n>>(~n&n-1).bit_length()).values())<<1 # Chai Wah Wu, Nov 24 2023

Crossrefs

Inverse Möbius transform of the characteristic function (1-A354108) of the complement of A354109.

Cf. A000265, A001222, A067019 (positions of odd terms), A353558 (parity of terms).

Sequence in context: A374366 A229893 A317683 * A198727 A294508 A035152

Adjacent sequences: A366368 A366369 A366370 * A366372 A366373 A366374

Keyword

nonn

Author

Antti Karttunen, Nov 23 2023

Status

approved