A345308 - OEIS

A345308

Decimal expansion of Sum_{p primes} log(p) / (p-1)^2.

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

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OFFSET

1,2

COMMENTS

Measures the excess divisibility of shifted primes. By Dirichlet's theorem, the density of primes q with p^j | q-1 is 1/phi(p^j), so the mean p-adic valuation of q-1 over primes q is Sum_{j>=1} 1/phi(p^j) = p/(p-1)^2. For a random integer that mean is 1/(p-1). The excess is exactly 1/(p-1)^2, and weighting it by log(p) and summing over p gives this constant c. Equivalently, the expected logarithm of the y-smooth part of q-1 exceeds that of a random integer by Sum_{p<=y} log(p)/(p-1)^2, which converges to c, so on the geometric-mean scale that smooth part is larger by the factor exp(c) = 3.410874826158870367... . Over half of c comes from p = 2 alone, where the mean 2-adic valuation of q-1 is 2 rather than 1. - James Conor Grogan, Aug 10 2026

LINKS

Table of n, a(n) for n=1..105.

Paul Erdős, Carl Pomerance and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica, Vol. 58, No. 4 (1991), pp. 363-385; alternative link.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C10 + 1).

Florian Luca and Igor E. Shparlinski, Average multiplicative orders of elements modulo n, Acta Arithmetica, Vol. 109, No. 4 (2003), pp. 387-411.

Eric Weisstein's World of Mathematics, Prime Factor, formula (13)-(14), (constant U).

FORMULA

Equals -Sum_{k>=2} (k-1)*P'(k), where P(s) is the prime zeta function. - Amiram Eldar, Nov 04 2025

Equals Sum_{p prime} log(p) * (Sum_{j>=1} 1/phi(p^j) - Sum_{j>=1} 1/p^j). - James Conor Grogan, Aug 10 2026

EXAMPLE

1.226968805653470005965662568745762562988257454901426311714794620109...

MATHEMATICA

ratfun = 1/((p - 1)^2); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 1000, 5000, 1000}]

CROSSREFS

Cf. A138312, A306016, A345364.

Sequence in context: A388701 A242978 A392562 * A231137 A371400 A188808

Adjacent sequences: A345305 A345306 A345307 * A345309 A345310 A345311

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Jun 13 2021

STATUS

approved