A388069 Decimal expansion of the asymptotic density of the integers k such that k and k+1 are both numbers whose number of divisors is a power of 2 (A372690).

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 [math.NT], 2025. See p. 7.

Formula

Equals Product_{p prime} (1 - 2/p^2 + Sum_{j>=2} (2/p^(2^j-1) - 2/p^(2^j))) (Tao and Teräväinen, 2025).

Example

0.44446069678909000638547564553450536620505334196836...

Prog

(PARI) my(jmax = 2, c1 = 0, c2 = 1); while(c2 != c1, c1 = c2; c2 = prodeulerrat(1 - 2/p^2 + sum(j = 2, jmax, 2/p^(2^j - 1) - 2/p^(2^j))); jmax++); c2

Crossrefs

Cf. A065474, A327839, A372690.

Sequence in context: A140744 A179414 A361248 * A139324 A111655 A175961

Adjacent sequences: A388066 A388067 A388068 * A388070 A388071 A388072

Keyword

nonn,cons

Author

Amiram Eldar, Dec 07 2025

Status

approved