A365870 Numbers k such that k and k+1 both have an even exponent of least prime factor in their prime factorization.

44, 48, 63, 80, 99, 116, 171, 175, 207, 260, 275, 315, 324, 332, 368, 387, 404, 475, 476, 495, 528, 531, 539, 548, 575, 603, 624, 636, 656, 692, 724, 747, 764, 819, 832, 891, 908, 924, 931, 960, 963, 980, 1024, 1035, 1052, 1071, 1075, 1088, 1124, 1179, 1196, 1232

Offset: 1

Author

Amiram Eldar, Sep 21 2023

Keywords

• nonn
• easy

Comments

Numbers k such that k and k+1 are both terms of A365869.

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 5, 42, 414, 4173, 41927, 419597, 4196917, 41972747, 419738185, 4197406018, ... . Apparently, the asymptotic density of this sequence exists and equals 0.04197... .

Examples

			44 is a term since the exponent of the prime factor 2 in the factorization 44 = 2^2 * 11 is 2, which is even, and the exponent of the prime factor 3 in the factorization 45 = 3^2 * 5 is also 2, which is even.
		

Links

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

Crossrefs

Subsequence of A365864 and A365869.

A365871 is a subsequence.

Programs

Mathematica

q[n_] := EvenQ[FactorInteger[n][[1, -1]]]; consec[kmax_] := Module[{m = 1, c = Table[False, {2}], s = {}}, Do[c = Join[Rest[c], {q[k]}]; If[And @@ c, AppendTo[s, k - 1]], {k, 1, kmax}]; s]; consec[1250]

PARI

lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = !(factor(k)[1,2]%2); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}