A334210 a(n) = sigma(prime(n) + 1) - sigma(prime(n)).

1, 3, 6, 7, 16, 10, 21, 22, 36, 42, 31, 22, 54, 40, 76, 66, 108, 34, 58, 123, 40, 106, 140, 144, 73, 114, 106, 172, 106, 126, 127, 204, 150, 196, 222, 148, 82, 130, 312, 186, 366, 154, 316, 100, 270, 265, 166, 280, 332, 202, 312, 504, 157, 476, 270, 456, 450, 286, 142, 294

Offset: 1

Author

Bernard Schott, Apr 18 2020

Keywords

• nonn

Comments

Lim_{n->oo} a(n) = oo because a(n) > sqrt(prime(n)) [see the reference], but this sequence is not monotone increasing.

a(n) is the sum of aliquot parts of the sum of divisors of n-th prime (see Marcus's formula). - Omar E. Pol, Apr 18 2020

Examples

			As prime(6) = 13, a(6) = sigma(14) - sigma(13) = 24 - 14 = 10.
		

References

• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 617 pp. 82, 280, Ellipses, Paris, 2004.

Crossrefs

Cf. A000203, A001065, A008333, A008864, A051027.

Programs

Maple

G:= seq(sigma(ithprime(p)+1)-sigma(ithprime(p)), p=1..200);

Mathematica

(DivisorSigma[1, # + 1] - # - 1)& @ Select[Range[300], PrimeQ] (* Amiram Eldar, Apr 18 2020 *)

PARI

a(n) = my(p=prime(n)); sigma(p+1) - (p+1); \\ Michel Marcus, Apr 18 2020

Formula

a(n) = A008333(n) - A008864(n).

From Michel Marcus, Apr 18 2020: (Start)

a(n) = A001065(A008864(n)).

a(n) = A051027(prime(n)) - A000203(prime(n)). (End)