A262153 Decimal expansion of Sum_{p prime} 1/(2^p-1), a prime equivalent of the Erdős-Borwein constant.

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

Offset

0,1

Comments

Erdős was interested in the question whether this constant is irrational. - Amiram Eldar, Apr 30 2020

Pratt gives a conditional proof that this constant is irrational. - Charles R Greathouse IV, Sep 26 2024

References

Paul Erdős, Some of my favourite unsolved problems, in A. Baker, B. Bollobás and A. Hajnal (eds.), A Tribute to Paul Erdős, Cambridge University Press, 1990, p. 470.

Links

Table of n, a(n) for n=0..99.

Thomas Bloom, Problem 69, Erdős Problems.

Paul Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.), (1948) Vol. 12, 63-66.

Paul Erdős, On the irrationality of certain series, Math. Student (1969) Vol. 36, 222-226.

Erdős problems database contributors, Erdős problem database, see no. 69.

Kyle Pratt, The irrationality of a prime factor series under a prime tuples conjecture, arXiv:2409.15185 [math.NT], 2024.

Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 [math.NT], 2025. See p. 4.

Eric Weisstein's World of Mathematics, Erdős-Borwein Constant.

Eric Weisstein's World of Mathematics, Prime Constant.

Formula

Equals Sum_{i>=1} 1/A001348(i). - R. J. Mathar, Feb 17 2016

Equals Sum_{k>=1} omega(k)/2^k, where omega(k) is the number of distinct primes dividing k (A001221). - Amiram Eldar, Apr 30 2020

Example

0.51694281980564038424051660847985627797854694791309124165028...

Mathematica

digits = 100; m0 = 100; dm = 100; s[m_] := s[m] = N[Sum[1/(2^Prime[n]-1), {n, 1, m}], digits+10]; s[m = m0]; Print[{m, s[m]}]; s[m = m + dm]; While[ Print[{m, s[m]}]; RealDigits[ s[m], 10, digits+5] != RealDigits[ s[m - dm], 10, digits+5], m = m + dm]; RealDigits[ s[m], 10, digits] // First

Prog

(PARI) suminf(k=1, omega(k)/2^k) \\ Michel Marcus, Apr 30 2020
(PARI) s=0.; forprime(p=2, bitprecision(1.)+1, s+=1./(2^p-1)); s \\ Charles R Greathouse IV, Sep 26 2024

Crossrefs

Cf. A051006, A065442.

Sequence in context: A306700 A058651 A164105 * A160824 A193586 A007397

Adjacent sequences: A262150 A262151 A262152 * A262154 A262155 A262156

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 13 2015

Status

approved