A372690 - OEIS

%I #7 May 10 2024 11:12:36

%S 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 54,55,56,57,58,61,65,66,69,70,73,77,78,82,85,86,87,88,93,94,101,102,

%U 103,104,105,106,109,110,113,114,118,119,122,127,128,129,130,133

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

%C 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 Amiram Eldar, <a href="/A372690/b372690.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 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 pow2Q[n_] := n == 2^IntegerExponent[n, 2]; q[n_] := q[n] = pow2Q[DivisorSigma[0, n]]; Select[Range[150], q[#] && q[# + 1] &]

%o (PARI) is(n) = {my(d = numdiv(n)); d >> valuation(d, 2) == 1;}

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

%Y Subsequence of A007674 and A036537.

%Y A372691 is a subsequence.

%Y Cf. A005237, A327839.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, May 10 2024