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

%I #6 May 10 2024 11:12:46

%S 135,296,343,375,999,1160,1431,1592,1624,2295,2375,2456,2727,2943,

%T 3429,3591,3624,3752,3992,4023,4184,4887,4913,5048,5144,5319,5480,

%U 6183,6344,6375,6858,7479,7624,7640,7749,7911,8072,8375,8936,9207,9368,9624,9855,10071,10232

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

%C First differs from A176313 at n = 14.

%C 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... .

%H Amiram Eldar, <a href="/A372691/b372691.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 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.

%t 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] &]

%o (PARI) is(n) = {my(f = factor(n), d = numdiv(f)); n > 1 && vecmax(f[, 2]) > 1 && 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 A013929, A068781, A175496 and A372690.

%Y Cf. A176313.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, May 10 2024