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

1,1

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... .

Links

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

Example

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.

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]

Prog

(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); }

Crossrefs

Subsequence of A365864 and A365869.

A365871 is a subsequence.

Sequence in context: A182261 A061757 A342187 * A088066 A116355 A071318

Adjacent sequences: A365867 A365868 A365869 * A365871 A365872 A365873

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved