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

Author

Jean-Marc Rebert, May 09 2025

Keywords

• nonn
• more

Comments

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)

Examples

			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.
		

Crossrefs

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

Programs

PARI

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

Formula

a(n) <= A317681(n).

Extensions

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