A391397 Numbers k such that d(k+1)/d(k) = 2^m for some integer m (positive, zero, or negative), where d(k) is the number of divisors of k.

1, 2, 5, 6, 7, 10, 13, 14, 21, 22, 23, 26, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 44, 46, 49, 53, 54, 55, 56, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 87, 88, 93, 94, 98, 101, 102, 103, 104, 105, 106, 109, 110, 113, 114, 116, 118, 119, 122, 127, 128

Offset

1,2

Comments

Tao and Teräväinen (2025) give 0.4888 as an empirical estimation for the asymptotic density of this sequence. They give the lower bound 0.48804, which is the approximate sum of the densities of the two disjoint subsequences A372690 and A391396.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

Example

1 is a term since d(2)/d(1) = 2/1 = 2^1.
2 is a term since d(3)/d(2) = 2/2 = 2^0.
6 is a term since d(7)/d(6) = 2/4 = 2^(-1).

Maple

dd:= map(numtheory:-tau, [$1..1000]):
r:= zip(`/`, dd[2..-1], dd[1..-2]):
select(t -> r[t] = 2^padic:-ordp(r[t], 2), [$1..999]); # Robert Israel, Dec 08 2025

Mathematica

d[n_] := d[n] = DivisorSigma[0, n]; q[n_] := Module[{r = d[n+1] / d[n]}, If[r >= 1, IntegerQ[r] && r == 2^IntegerExponent[r, 2], IntegerQ[1/r] && 1/r == 2^IntegerExponent[1/r, 2]]]; Select[Range[200], q]

Prog

(PARI) is(r) = r >= 1 && denominator(r) == 1 && r >> valuation(r, 2) == 1;
list(kmax) = {my(d1 = numdiv(1), d2); for(k = 2, kmax+1, d2 = numdiv(k); if(is(d2/d1) || is(d1/d2), print1(k-1, ", ")); d1 = d2); }

Crossrefs

Cf. A000005, A036537, A377562.

Subsequences: A372690, A391396.

Sequence in context: A047578 A259605 A367695 * A372690 A284393 A287366

Adjacent sequences: A391394 A391395 A391396 * A391398 A391399 A391400

Keyword

nonn,easy

Author

Amiram Eldar, Dec 08 2025

Status

approved