A372691 Numbers k such that k and k+1 are both nonsquarefree numbers whose number of divisors is a power of 2 (A175496).

135, 296, 343, 375, 999, 1160, 1431, 1592, 1624, 2295, 2375, 2456, 2727, 2943, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4887, 4913, 5048, 5144, 5319, 5480, 6183, 6344, 6375, 6858, 7479, 7624, 7640, 7749, 7911, 8072, 8375, 8936, 9207, 9368, 9624, 9855, 10071, 10232

Offset: 1

Author

Amiram Eldar, May 10 2024

Keywords

• nonn
• easy

Comments

First differs from A176313 at n = 14.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 0, 5, 43, 404, 4086, 40839, 408366, 4083039, 40830831, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004083... .

Examples

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17 are both nonsquarefree numbers, and the number of divisors of 135 and 136 are both 8 = 2^3.
343 is a term since 343 = 7^3 and 344 = 2^3 * 43 are both nonsquarefree numbers, the number of divisors of 343 is 4 = 2^2, and the number of divisors of 344 is 8 = 2^3.
		

Links

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

Crossrefs

Subsequence of A013929, A068781, A175496 and A372690.

Cf. A176313.

Programs

Mathematica

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; q[n_] := q[n] = Module[{e = FactorInteger[n][[;;, 2]]}, Max[e] > 1 && pow2Q[Times @@ (e+1)]]; Select[Range[500], q[#] && q[# + 1] &]

PARI

is(n) = {my(f = factor(n), d = numdiv(f)); n > 1 && vecmax(f[, 2]) > 1 && 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);}