A256936 Decimal expansion of Sum_{k>=1} phi(k)/2^k, where phi is Euler's totient function.

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

Offset

1,2

Comments

Equals p + 1/2, where p is the asymptotic probability that two positive even integers selected independently at random have coprime 2-adic valuations (A007814). - Amiram Eldar, Mar 15 2026

References

Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, p. 139.

Links

Table of n, a(n) for n=1..99.

Thomas Bloom, Problem 249, Erdős Problems.

Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique, L'enseignement Mathématique, Université de Genève, 1980, p. 61.

Erdős problems database contributors, Erdős problem database, see no. 249.

Eric Weisstein's MathWorld, Totient Function.

Wikipedia, Euler's totient function.

Formula

Equals Sum_{k>=1} A007431(k)/(2^k - 1). - Amiram Eldar, Jun 23 2020

Equals 1/2 + Sum_{k>=1} mu(k)/(2^k-1)^2, where mu is the Moebius function (A008683). - Amiram Eldar, Mar 15 2026

Example

1.36763080198502235079050814621308813907489199896...

Mathematica

digits = 99; m0 = 10; dd = 10; Clear[f]; f[m_] := f[m] = Sum[EulerPhi[n]/2^n, {n, 1, m}] // N[#, digits + 2*dd]&; f[m = m0] ; While[RealDigits[f[2*m], 10, digits + dd ] != RealDigits[f[m], 10, digits + dd ], m = 2*m; Print[m]]; RealDigits[f[m], 10, digits] // First

Prog

(PARI) suminf(n=1, eulerphi(n)/2^n) \\ Charles R Greathouse IV, Apr 20 2016

Crossrefs

Cf. A000010, A007431, A007814, A008683.

Sequence in context: A156648 A278688 A016616 * A021276 A329516 A290943

Adjacent sequences: A256933 A256934 A256935 * A256937 A256938 A256939

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 13 2015

Status

approved