A072815 Sum of proper divisors of 6n + 1.

0, 1, 1, 1, 6, 1, 1, 1, 8, 17, 1, 1, 1, 1, 23, 21, 1, 1, 1, 29, 12, 1, 27, 1, 35, 1, 1, 1, 14, 73, 1, 29, 1, 1, 47, 1, 39, 1, 1, 53, 1, 33, 35, 45, 59, 1, 1, 1, 18, 65, 51, 1, 1, 41, 109, 1, 1, 57, 1, 77, 20, 1, 1, 1, 191, 41, 1, 45, 1, 89, 1, 69, 1, 1, 95, 53, 1

Offset: 0

Author

Hisanobu Shinya (ilikemathematics(AT)hotmail.com), Jul 14 2002

Keywords

• nonn

Comments

The square root of t(n) < s(t(4n-1, 4n-2, 4n-3, 4n-4)) < s(t(4n)).

Examples

			a(1) = s(t(1)) = 1 since t(1) = 7 and s(7) = 1 under the definition of the restricted divisor function.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Restricted Divisor Function.

Crossrefs

Cf. A001065, A016921, A145426, A363031.

Programs

Mathematica

Table[c=6n+1; DivisorSigma[1,c]-c, {n,0,80}] (* Harvey P. Dale, Nov 13 2013 *)

PARI

a(n) = sigma(6*n + 1) - 6*n - 1; \\ Amiram Eldar, Apr 12 2024

Formula

a(n) = s(t(n)), where t(n) = 6n + 1 and s(n) is the restricted divisor function.

From Amiram Eldar, Apr 12 2024: (Start)

a(n) = A363031(n) - A016921(n) = A001065(A016921(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = P^2/3 - 3 = A145426 = 0.289868... . (End)

Extensions

Corrected by Harvey P. Dale, Nov 13 2013