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A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

%I A382960 #22 Apr 19 2025 18:06:51
%S A382960 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A382960 27,28,29,30,31,32,33,34,35,36,38,39,40,42,44,45,46,48,50,51,52,54,56,
%U A382960 57,58,60,62,63,64,66,68,69,70,72,74,75,76,78,80,81,82,84
%N A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).
%C A382960 Numbers k whose reduced residue system does not intersect A286708 (i.e., powerful numbers that are not prime powers).
%C A382960 Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q^2 = A382767(k). Then this sequence is that of k such that k < m.
%C A382960 This sequence is finite following arguments akin to those in A051250 and A382659, with 626 terms.
%C A382960 Sequences A048597, A051250, and A382659 are proper subsets of this sequence.
%H A382960 Michael De Vlieger, <a href="/A382960/b382960.txt">Table of n, a(n) for n = 1..626</a>
%e A382960 Let omega = A001221.
%e A382960 For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = (2*3)^2 = 36.
%e A382960 For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 64, 81, 128}.
%e A382960   31 is in the sequence since 31 < m, m = (2*3)^2 = 36, but 37 is not a term since 37 > 36.
%e A382960   25 is in the sequence since 25 < m, m = 36.
%e A382960   49 is not a term since 49 > 36, and 243 is not a term since 243 > 100, 100 = (2*5)^2, etc.
%e A382960 For omega = 2, we have the squarefree numbers {6, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218}.
%e A382960   Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, .., 1152}, since m = (5*7)^2 = 1225.
%e A382960   Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, ..., 400}, since m = (3*7)^2 = 441.
%e A382960   Intersection with A033847 = {k : rad(k) = 14} is {14, 28, 56, ..., 224}, since m = (3*5)^2 = 225.
%e A382960   Intersection with A033848 = {k : rad(k) = 15} is {15, 45, 75, 135}, since m = (2*7)^2 = 196, etc.
%t A382960 Select[Range[510510], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++]][[-1, 1]] ] ]
%Y A382960 Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A286708, A380539, A382659 (k such that k < p^2*q), A382767.
%K A382960 nonn,easy,fini,full
%O A382960 1,2
%A A382960 _Michael De Vlieger_, Apr 14 2025