A335118 Decimal expansion of the sum of the reciprocals of the perfect numbers.

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

Offset: 0

Author

Amiram Eldar, May 24 2020

Keywords

• nonn
• cons

Comments

Bayless and Klyve (2013) calculated the first 149 terms of this sequence. The terms beyond this are uncertain due to the possible existence of odd perfect numbers larger than 10^300.

Examples

			0.20452014283892643017813442909845557667731148935076...
		

References

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 244.

Links

• Jonathan Bayless and Dominic Klyve, Reciprocal sums as a knowledge metric: theory, computation, and perfect numbers, The American Mathematical Monthly, Vol. 120, No. 9 (2013), pp. 822-831, alternative link, preprint.

Crossrefs

Cf. A000396, A173898.

Programs

Mathematica

RealDigits[Sum[1/2^(p - 1)/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]]
RealDigits[Total[1/PerfectNumber[Range[15]]],10,120][[1]] (* Harvey P. Dale, Nov 25 2023 *)

Formula

Equals Sum_{k>=1} 1/A000396(k).