A383920 -id:A383920 - cp's OEIS frontend

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

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

A383921 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, 91, 841, 13861, 360361

Offset: 2

Michel Marcus, May 15 2025

The corresponding x are: 0, 0, 0, 0, 89, 839, 13859, 360359, ...
a(10) <= 61261201 via sigma(61261201 - 61261199) + sigma(61261201 + 61261199) >= 10*61261201. - David A. Corneth, May 16 2025
a(11) <= 20951330401 via sigma(20951330401 - 20951330399) + sigma(20951330401 + 20951330399) >= 11*20951330401. - Michel Marcus, May 22 2025

Cf. A000203, A004394, A383758, A383920.

isok(k,n) = my(v=vector(k, x, (sigma(k - x + 1) + sigma(k + x - 1))/k)); my(vk = select(t->(t>=n), v, 1)); if (#vk != 0, v[1]);
a(n) = my(k=1); while (!isok(k, n), k++); k;

a(9) from Jean-Marc Rebert, May 16 2025

cp's OEIS Frontend

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

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