A392467 Numbers whose sum of odd indexed divisors divides the sum of its divisors.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 98

Offset

1,2

Comments

Numbers k such that A086988(k) divides A086989(k).

In the sequences A086988 and A086989, the divisors of k are listed in increasing order and the indices range from 1 to tau(k).

For any positive integer k, define r(k):=sigma_e(k)/sigma_o(k), where sigma_e=A086989 and sigma_o=A086988, and we call k a r-index ratio number. This sequence gives those numbers where this ratio is an integer.

Links

Table of n, a(n) for n=1..68.

B Mittou, A New Classification of Positive Integers Via New Divisor Functions, arXiv:2509.08844 [math.GM], 2025.

Maple

filter:= proc(n) local S, i;
  S:= sort(convert(NumberTheory:-Divisors(n), list));
  add(S[i], i=2..nops(S), 2) mod add(S[i], i=1..nops(S), 2) = 0
end proc:
select(filter, [$1..100]); # Robert Israel, Feb 22 2026

Mathematica

q[k_] := Divisible @@ Reverse[Total /@ Transpose[Partition[Divisors[k], 2, 2, {1, 1}, {0}]]]; Select[Range[100], q] (* Amiram Eldar, Feb 20 2026 *)

Prog

(PARI) isok(k)=my(se=0, so=0, i=0); fordiv(k, d, i=!i; if(i, so+=d, se+=d)); se%so==0; \\ Andrew Howroyd, Feb 19 2026
(Python)
from sympy import divisors
def ok(n): return sum((divs:=divisors(n))[1::2])%sum(divs[::2]) == 0
print([k for k in range(1, 101) if ok(k)]) # Michael S. Branicky, Feb 22 2026

Crossrefs

Cf. A000203, A086988, A086989.

Includes A006881 (squarefree semiprimes) and A246551 (primes to odd powers).

Sequence in context: A109423 A028801 A348751 * A348738 A239161 A162644

Adjacent sequences: A392464 A392465 A392466 * A392468 A392469 A392470

Keyword

nonn

Author

B Mittou, Feb 19 2026

Status

approved