cp's OEIS Frontend

A372690 Numbers k such that k and k+1 are both numbers whose number of divisors is a power of 2 (A036537).

%I A372690 #9 Apr 10 2025 08:17:18
%S A372690 1,2,5,6,7,10,13,14,21,22,23,26,29,30,33,34,37,38,39,40,41,42,46,53,
%T A372690 54,55,56,57,58,61,65,66,69,70,73,77,78,82,85,86,87,88,93,94,101,102,
%U A372690 103,104,105,106,109,110,113,114,118,119,122,127,128,129,130,133
%N A372690 Numbers k such that k and k+1 are both numbers whose number of divisors is a power of 2 (A036537).
%C A372690 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 6, 44, 449, 4450, 44462, 444471, 4444647, 44446255, 444461038, 4444607360, ... . Apparently, the asymptotic density of this sequence exists and equals 0.44446... .
%H A372690 Amiram Eldar, <a href="/A372690/b372690.txt">Table of n, a(n) for n = 1..10000</a>
%e A372690 1 is a term since the number of divisors of 1 is 1 = 2^0, and the number of divisors of 1 + 1 = 2 is 2 = 2^1.
%e A372690 54 is a term since the number of divisors of 54 is 8 = 2^3, and the number of divisors of 54 + 1 = 55 is 4 = 2^2.
%t A372690 pow2Q[n_] := n == 2^IntegerExponent[n, 2]; q[n_] := q[n] = pow2Q[DivisorSigma[0, n]]; Select[Range[150], q[#] && q[# + 1] &]
%o A372690 (PARI) is(n) = {my(d = numdiv(n)); d >> valuation(d, 2) == 1;}
%o A372690 lista(kmax) = {my(is1 = is(1), is2); for(k = 2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}
%Y A372690 Subsequence of A007674 and A036537.
%Y A372690 A372691 is a subsequence.
%Y A372690 Cf. A005237, A327839.
%K A372690 nonn,easy
%O A372690 1,2
%A A372690 _Amiram Eldar_, May 10 2024