A338475 Decimal expansion of the sum of reciprocals of the smallest primes > 2^k for k >= 0.

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

Offset: 1

Author

Bernard Schott, Oct 29 2020

Keywords

• nonn
• cons

Comments

If q(k) = A014210(k) is the smallest prime > 2^k, then 2^k < q(k), so Sum_{k>=0} 1/q(k) < Sum_{k>=0} 1/2^k = 2; hence, the sum of the reciprocals of these primes q(k) form a convergent series.

Examples

			1.2404071466559606289464180214057283392313810734691...
		

References

• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 615 pp. 82 and 279, Ellipses, Paris, 2004. Warning : gives Sum_{k>=1} 1/A104080(k) = 0.7404...

Crossrefs

Cf. A014210, A104080, A203074.

Programs

Maple

evalf(sum(1/nextprime(2^k), k=0..infinity),90);

Mathematica

ndigits = 90; RealDigits[Sum[1/NextPrime[2^k], {k, 0, ndigits/Log10[2] + 1}], 10, ndigits][[1]] (* Amiram Eldar, Oct 29 2020 *)

PARI

suminf(k=0, 1/nextprime(2^k+1)) \\ Michel Marcus, Oct 29 2020

Formula

Equals Sum_{k>=0} 1/A014210(k).