A365864 -id:A365864 - cp's OEIS frontend

A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.

423, 475, 1323, 1375, 1519, 2007, 2223, 2275, 2871, 3123, 3175, 3211, 3283, 3479, 3575, 3751, 3771, 4023, 4075, 4475, 4923, 4959, 4975, 5047, 5535, 5823, 5875, 6723, 6775, 6811, 7299, 7623, 7675, 8107, 8379, 8523, 8575, 8955, 9423, 9475, 10323, 10339, 10375, 10467

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that k, k+1 and k+2 are all terms of A283050.
Numbers of the form 4*k+2 are not terms of A283050. Therefore, there are no runs of 4 or more consecutive integers, and all the terms of this sequence are of the form 4*k+3.
The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 40, 429, 4419, 44352, 444053, 4441769, 44421000, 444220814, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004442... .

			423 is a term since 3 is the least prime factor of 423 and 423 is divisible by 3^2 = 9, 2 is the least prime factor of 424 and 424 is divisible by 2^2 = 4, and 5 is the least prime factor of 425 and 425 is divisible by 5^2 = 25.

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

Cf. A067029.

Subsequence of A004767, A070258, A283050 and A365864.

Select[4 * Range[2700] + 3, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 2 &] &]
SequencePosition[Table[If[Divisible[n,FactorInteger[n][[1,1]]^2],1,0],{n,11000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Aug 05 2024 *)

is(n) = factor(n)[1,2] >= 2;
lista(kmax) = forstep(k = 3, kmax, 4, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

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

Original entry on oeis.org

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

Amiram Eldar, Sep 21 2023

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... .

			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.

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

Subsequence of A365864 and A365869.

A365871 is a subsequence.

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]

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

cp's OEIS Frontend

A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.

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