A393444 Decimal expansion of Sum_{n >= 1} 1/A171397(prime(n)).

3, 0, 8, 7, 8

Offset

1,1

Comments

Start with the divergent series Sum_{n >= 1} 1/prime(n) and make it converge by replacing 1/prime(n) by 1/U(prime(n)), where U(k) means read k in base 10 and write it in base 11 (cf. A171397). The sum becomes 1/2 + 1/3 +1/5 + 1/7 + 1/12 + 1/14 + 1/18 + 1/20 + 1/25 + ..., which converges, albeit rather slowly.

Before Victor Miller's work, not a single digit of the value of the sum was known for certain. Hans Havermann had found that even after 10^12 terms the sum was about 2.94 and the leading digit was still uncertain (see "A Nasty Surprise ...").

Links

Table of n, a(n) for n=1..5.

Victor Miller, A Difficult Prime Series Problem, Talk in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, Apr 02 2026 (Vimeo); Slides.

N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence in slides 13-27]

Doron Zeilberger, Homepage for Experimental Mathematics Seminar.

Example

The sum is 3.0878... Victor Miller shows that the value lies in the interval (3.0878725218714846, 3.0878855128344616).

Crossrefs

Cf. A171397, A375805.

Sequence in context: A107370 A201580 A234518 * A068458 A238258 A011082

Adjacent sequences: A393441 A393442 A393443 * A393445 A393446 A393447

Keyword

nonn,cons,base,more

Author

N. J. A. Sloane, Apr 02 2026

Status

approved