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

1,2

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.

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...

Links

Table of n, a(n) for n=1..90.

Formula

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

Example

1.2404071466559606289464180214057283392313810734691...

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 *)

Prog

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

Crossrefs

Cf. A014210, A104080, A203074.

Sequence in context: A011994 A258773 A054003 * A134352 A152648 A327898

Adjacent sequences: A338472 A338473 A338474 * A338476 A338477 A338478

Keyword

nonn,cons

Author

Bernard Schott, Oct 29 2020

Status

approved