A391684 Number of divisors d of n which also divide (n/d)^2 - 1.

1, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 3, 3, 2, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 2, 6, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 2, 6, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 2, 6, 2, 3, 2, 2, 2, 5, 2, 3, 3, 4, 2, 4, 2, 3, 3, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 3, 3, 2, 5, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3

Offset

1,2

Comments

From Robert Israel, Mar 29 2026: (Start)

a(n) >= 2 for n > 1, since d = 1 and d = n always work.

a(n) = 2 if n is a prime power (A246655). (End)

Links

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

Example

a(4) = 2 because (4/d)^2 - 1 == 0 (mod d) for d = 1 (or 4) of n = 4, but d = 2 is a nondivisor of 3 = (4/2)^2 - 1.

Maple

f:= proc(n)
  nops(select(d -> (n/d)^2-1 mod d = 0, NumberTheory:-Divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 29 2026

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[(n/#)^2 - 1, #] &]; Array[a, 100] (* Amiram Eldar, Mar 29 2026 *)

Prog

(Magma) [#[d: d in Divisors(n) | ((n div d)^2 - 1) mod d eq 0]: n in [1..100]];
(PARI) a(n) = sumdiv(n, d, (((n/d)^2-1) % d) == 0); \\ Michel Marcus, Apr 06 2026

Crossrefs

Cf. A000005, A034444, A116974, A163671, A246655, A394362.

Sequence in context: A216321 A058263 A384834 * A232398 A389781 A048669

Adjacent sequences: A391681 A391682 A391683 * A391685 A391686 A391687

Keyword

nonn

Author

Juri-Stepan Gerasimov, Mar 28 2026

Status

approved