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

1, 2, 5, 6, 7, 10, 13, 14, 21, 22, 23, 26, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 46, 53, 54, 55, 56, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 87, 88, 93, 94, 101, 102, 103, 104, 105, 106, 109, 110, 113, 114, 118, 119, 122, 127, 128, 129, 130, 133

Offset: 1

Author

Amiram Eldar, May 10 2024

Keywords

• nonn
• easy

Comments

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... .

Examples

			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.
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.
		

Links

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

Crossrefs

Subsequence of A007674 and A036537.

A372691 is a subsequence.

Cf. A005237, A327839.

Programs

Mathematica

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

PARI

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