A391396 - OEIS

%I #11 Dec 14 2025 01:50:17

%S 44,49,75,98,116,147,171,242,243,244,260,275,315,332,360,363,387,475,

%T 476,507,524,531,539,548,549,603,604,636,692,724,725,735,747,764,774,

%U 819,844,845,846,867,908,924,927,931,963,1035,1075,1083,1124,1175,1179,1183

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

%C 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).

%H Amiram Eldar, <a href="/A391396/b391396.txt">Table of n, a(n) for n = 1..10000</a>

%H Terence Tao and Joni Teräväinen, <a href="https://arxiv.org/abs/2512.01739">Quantitative correlations and some problems on prime factors of consecutive integers</a>, arXiv:2512.01739 [math.NT], 2025. See p. 7.

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

%o (PARI) is(k) = {my(d = numdiv(k)); d >> valuation(d, 2) == 3;}

%o list(kmax) = {my(is1 = is(1), is2); for(k = 2, kmax+1, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}

%Y Cf. A372690, A388069.

%Y Subsequence of A377562 and A391397.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Dec 08 2025