A259931 Decimal expansion of the sum of the reciprocals of the averages of adjacent pairs of perfect numbers (A000396).

0, 6, 2, 8, 7, 2, 2, 9, 4, 0, 9, 4, 1, 9, 7, 0, 1, 4, 8, 9, 9, 7, 6, 9, 1, 8, 9, 3, 0, 8, 7, 5, 0, 6, 2, 6, 6, 1, 6, 0, 3, 2, 7, 8, 9, 3, 1, 9, 9, 4, 8, 0, 4, 3, 8, 2, 1, 3, 1, 0, 5, 0, 8, 6, 5, 9, 6, 8, 8, 8, 4, 7, 1, 2, 3, 5, 8, 5, 7, 2, 1, 4, 9, 7, 5, 5, 2, 9, 5, 0, 0, 7, 7, 1, 0, 4, 3, 0, 7, 7, 8, 4, 1, 2, 0, 0

Offset: 0

Author

José de Jesús Camacho Medina, Aug 17 2015

Keywords

• nonn
• cons

Examples

			=0.0628722940941970148997691893087506266160327893199480438213105086596888471...
= 1/17 + 1/262 + 1/4312 + 1/16779232 + 1/4311709696 + 1/73014280192 + ...
		

Links

• Jonathan Bayless and Dominic Klyve, Reciprocal sums as a knowledge metric, Amer Math Monthly 120 (November, 2013) 822-831.
• Steven Finch, Amicable Pairs and Aliquot Sequences, Oct 31 2013. [Cached copy, with permission of the author]
• MathOverflow, Sum of the reciprocal of perfect numbers, Jun 10 2012.
• José Camacho Medina's Matematico Fresnillense, La Constante entre Numeros Perfectos (in Spanish).

Crossrefs

Cf. A000396.

Programs

Mathematica

exp = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521} (* see A000043 *); pn[k_] := 2^(exp[[k]] - 1)(2^exp[[k]] - 1); RealDigits[Sum[2/(pn[k] + pn[k + 1]), {k, 1, 12}], 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2015 *)

Formula

Equals Sum_{n>=1} 2/(A000396(n) + A000396(n+1)).

Equals Sum_{n>=1} 1/A259849(n).

Extensions

More terms from Jon E. Schoenfield, Aug 19 2015

More terms from Robert G. Wilson v, Dec 15 2015