A391396 Numbers k such that k and k+1 are both numbers whose number of divisors is 3 times a power of 2 (A377562).

44, 49, 75, 98, 116, 147, 171, 242, 243, 244, 260, 275, 315, 332, 360, 363, 387, 475, 476, 507, 524, 531, 539, 548, 549, 603, 604, 636, 692, 724, 725, 735, 747, 764, 774, 819, 844, 845, 846, 867, 908, 924, 927, 931, 963, 1035, 1075, 1083, 1124, 1175, 1179, 1183

Offset

1,1

Comments

The asymptotic density of this sequence is d * Sum_{primes p_0 != p_1} Product_{i=0..1} (Sum_{j>=0} (1/p_i^(3*2^j-1) - 2/p_i^(3*2^j)))/(1 - 2/p_i^2 + Sum_{j>=2} (2/p_i^(2^j-1) - 1/p_i^(2^j))) = 0.04358..., where d = A388069 (Tao and Teräväinen, 2025).

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.

Mathematica

pow23Q[n_] := n == 3 * 2^IntegerExponent[n, 2]; q[n_] := q[n] = pow23Q[DivisorSigma[0, n]]; Select[Range[1500], q[#] && q[# + 1] &]

Prog

(PARI) is(k) = {my(d = numdiv(k)); d >> valuation(d, 2) == 3; }
list(kmax) = {my(is1 = is(1), is2); for(k = 2, kmax+1, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2); }

Crossrefs

Cf. A372690, A388069.

Subsequence of A377562 and A391397.

Sequence in context: A088066 A116355 A071318 * A085651 A066124 A270298

Adjacent sequences: A391393 A391394 A391395 * A391397 A391398 A391399

Keyword

nonn,easy

Author

Amiram Eldar, Dec 08 2025

Status

approved