A390582 a(n) = (2*n - sigma(n))/gcd(n, sigma(n)).

1, 1, 2, 1, 4, 0, 6, 1, 5, 1, 10, -1, 12, 2, 2, 1, 16, -1, 18, -1, 10, 4, 22, -1, 19, 5, 14, 0, 28, -2, 30, 1, 6, 7, 22, -19, 36, 8, 22, -1, 40, -2, 42, 1, 4, 10, 46, -7, 41, 7, 10, 3, 52, -2, 38, -1, 34, 13, 58, -4, 60, 14, 22, 1, 46, -2, 66, 5, 14, -2, 70, -17

Offset

1,3

Comments

Empirical observation: mumbers in this sequence that are positive and close to 1 are more likely to produce regular amicable pairs under the Wiethaus's Rule (see Garcia et al).

The positions of -1's in the sequence are given by A153501.

The positions of 0's in the sequence are given by A000396.

The positions of 1's in the sequence are given by A271816.

If a(n) is negative, n is abundant (A005101).

If a(n) is positive, n is deficient (A005100).

If instead a(n) is 0, n is perfect (A000396).

Links

Hayden Chesnut, Table of n, a(n) for n = 1..10000

M. García, J.M. Pedersen and H.J.J. te Riele, Amicable pairs : a survey, Fields Institute Communications, 41 (2004), 1-19. See p. 193.

Formula

a(n) = A033879(n)/A009194(n).

a(prime(n)) = prime(n) - 1, and those are also the records. - Peter Luschny, Nov 13 2025

Example

For n = 1, (2*n - sigma(n)) = 1 and gcd(n,sigma(n)) = 1 so (2*n - sigma(n))/gcd(n, sigma(n)) = 1.
For n = 9, a(prime(9)) = 22 = prime(9) - 1;

Maple

a := proc(n) NumberTheory:-SumOfDivisors(n); (2*n - %)/igcd(n, %) end: seq(a(n), n = 1..72);  # Peter Luschny, Nov 11 2025

Mathematica

a[n_] := (2*n - #)/GCD[n, #]& @ DivisorSigma[1, n]; Array[a, 100] (* Amiram Eldar, Nov 11 2025 *)

Prog

(Python)
from sympy import divisor_sigma, gcd
lists = []
for A1 in range(1, 1000):
    A2 = divisor_sigma(A1)
    A3 = gcd(A1, A2)
    lists.append((A1*2-A2)/A3)
print(lists[:100])
(PARI) a(n) = my(s=sigma(n)); (2*n-s)/gcd(n, s); \\ Michel Marcus, Nov 11 2025

Crossrefs

Cf. A000203 (sigma), A000396, A005100, A005101, A009194, A033879, A153501, A271816.

Sequence in context: A326144 A120112 A233150 * A103977 A378600 A109883

Adjacent sequences: A390579 A390580 A390581 * A390583 A390584 A390585

Keyword

sign,easy

Author

Hayden Chesnut, Nov 11 2025

Status

approved