A383269 -id:A383269 - cp's OEIS frontend

A383268 Numbers k for which sigma(k - x) + sigma(k + x) = 4*k has at least one nonnegative solution.

6, 13, 15, 17, 28, 33, 39, 42, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 69, 71, 77, 78, 82, 89, 90, 93, 95, 99, 101, 107, 111, 115, 118, 120, 121, 123, 125, 129, 131, 139, 141, 149, 153, 161, 165, 167, 171, 177, 179, 182, 183, 190, 195, 196, 197, 201, 204, 213, 215

Offset: 1

Felix Huber, Apr 24 2025

Supersequence of A000396 because sigma(A000396(n) - x) + sigma(A000396(n) + x) = 4*A000396(n) has the solution x = 0.

			15 is in the sequence because sigma(15 - x) + sigma(15 + x) = 4*15 has the solution x = 5: sigma(15 - 5) + sigma(15 + 5) = sigma(10) + sigma(20) = 18 + 42 = 60 = 4*15.

Michel Marcus, Table of n, a(n) for n = 1..10000 (terms up to n=1000 by Felix Huber).
Eric Weisstein's World of Mathematics, Perfect Number

Cf. A000203, A000396, A186103, A383269.

with(NumberTheory):
A383268:=proc(N) # To get the first N terms.
    local k,x,K;
    K:=[];
    for k while nops(K)A383268(59);

isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 4*k, return(1))); \\ Michel Marcus, Apr 26 2025

A383758 Least integer k for which sigma(k - x) + sigma(k + x) = n*k has at least one solution.

Original entry on oeis.org

1, 2, 6, 24, 93, 1952, 14412, 361881, 61824672

Offset: 2

Jean-Marc Rebert, May 09 2025

The corresponding x are : 0, 0, 0, 0, 87, 1828, 13308, 358839, ...
a(10) <= 61824672 via sigma(61824672 - 60697728) + sigma(61824672 + 60697728) = 10*61824672. - Michel Marcus, May 20 2025
a(11) <= 43293761280 via sigma(43293761280 - 40511560320) + sigma(43293761280 + 40511560320) == 11*43293761280. - Michel Marcus, May 25 2025
Note that for n=2,3,4,5,8,and 9, we have k+x = A383920(n). - Michel Marcus, Jun 09 2025
From David A. Corneth, Jun 13 2025: (Start)
a(10) = 61824672. We must have sigma(k-x) >= 5*(k-x) or sigma(k+x) >= 5 * (k+x).
The numbers <= 2*61824672 that have this property are 122522400. It has been checked that if k + x = 122522400 then k must be 61824672 to get the smallest such k. (End)

			a(4) = 6 because the equation sigma(6-x) + sigma(6+x) = 4*6 has the solution x = 0 and no smaller number possesses this property. See A000396, A383268, and A383269.
a(5) = 24 because the equation sigma(24-x) + sigma(24+x) = 5*24 has the solution x = 0. This is verified as follows: sigma(24-0) + sigma(24+0) = sigma(24) + sigma(24) = 60 + 60 = 120 = 5*24. Moreover, no smaller number possesses this property. See A141643.
a(6) = 93 because the equation sigma(93 - x) + sigma(93 + x) = 6 * 93 has the solution x = 87: sigma(93 - 87) + sigma(93 + 87) = sigma(6) + sigma(180) = 12 + 546 = 6*93. Moreover, no smaller number possesses this property.

Cf. A000203, A000396, A141643, A317681, A383268, A383269.

isok(k,n) = forstep(x=k-1, 0, -1, if (sigma(k - x) + sigma(k + x) == n*k, return(1)));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 10 2025

a(n) <= A317681(n).

a(10) from Michel Marcus and David A. Corneth, Jun 13 2025

cp's OEIS Frontend

A383268 Numbers k for which sigma(k - x) + sigma(k + x) = 4*k has at least one nonnegative solution.

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