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
Examples
0.51694281980564038424051660847985627797854694791309124165028...
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
- Paul Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.), Vol. 12 (1948), pp. 63-66.
- Paul Erdős, On the irrationality of certain series, Math. Student, Vol. 36 (1969), pp. 222-226.
- Kyle Pratt, The irrationality of a prime factor series under a prime tuples conjecture, arXiv preprint (2024). arXiv:2409.15185 [math.NT]
- Eric Weisstein's World of Mathematics, Erdős-Borwein Constant.
- Eric Weisstein's World of Mathematics, Prime Constant.
Programs
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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 -
PARI
suminf(k=1, omega(k)/2^k) \\ Michel Marcus, Apr 30 2020
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PARI
s=0.; forprime(p=2,bitprecision(1.)+1,s+=1./(2^p-1)); s \\ Charles R Greathouse IV, Sep 26 2024
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
Comments