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A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

%I A382659 #17 Apr 19 2025 18:06:42
%S A382659 1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,20,22,24,26,28,30,32,34,36,38,
%T A382659 40,42,44,48,50,54,60,66,70,72,78,84,90,96,102,108,114,120,126,132,
%U A382659 138,144,150,156,162,168,174,180,210,240,252,270,300,330,360,390
%N A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).
%C A382659 Numbers k whose reduced residue system (RRS) does not intersect A126706 (i.e., the sequence of numbers that are neither squarefree nor prime powers). Alternatively, numbers k whose RRS is a subset of A303554 (i.e., the union of powers of primes and squarefree numbers).
%C A382659 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 = A382248(k). Then this sequence is that of k such that k < m.
%C A382659 There are 72 terms in this sequence.
%C A382659 Sequences A048597 and A051250 are proper subsets of this sequence.
%H A382659 Michael De Vlieger, <a href="/A382659/b382659.txt">Table of n, a(n) for n = 1..72</a>
%e A382659 Let omega = A001221.
%e A382659 For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = 2^2 * 3 = 12.
%e A382659 For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 16, 32}.
%e A382659   11 is in the sequence since 11 < m, m = 2^2 * 3 = 12, but 13 is not, since 13 > 12.
%e A382659   9 is in the sequence since 9 < m, m = 2^2 * 5 = 20.
%e A382659   25 is not a term since 25 > 12, and 27 is not a term since 27 > 20.
%e A382659 For omega = 2, we have the subset {6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 34, 36, 38, 40, 44, 48, 50, 54, 72, 96, 108, 144, 162}.
%e A382659   38 = 2*19 is a term since 38 < 45, 45 = 3^2 * 5, but 46 = 2*23 is not, since 46 > 45.
%e A382659   15 = 3*5 is a term since 15 < 20, but 21 is not, since 21 > 20 and 35 is not, since 35 > 12.
%e A382659   Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162}, since m = 5^2 * 7 = 175.
%e A382659   Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, 50}, since m = 3^2 * 7 = 63.
%e A382659   Intersection with A033847 = {k : rad(k) = 14} is {14, 28}, since m = 3^2 * 5 = 45, etc.
%e A382659 For omega = 3, we have the subset {30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 240, 252, 270, 300, 360, 450, 480}, of which {30, 42, 66, 70, 78, 102, 114, 138, 174} are squarefree.
%e A382659   Intersection with A143207 = {k : rad(k) = 30} is {30, 60, 90, .., 480} because m = 7^2 * 11 = 539.
%e A382659   Intersection with 42*A108319 = {k : rad(k) = 42} is {42, 84, 126, 168}, since m = 5^2 * 11 = 275, etc.
%e A382659 For omega = 4, we have the subset {210, 330, 390, 420, 510, 630, 840, 1050, 1260, 1470}, of which {210, 330, 390, 510} are squarefree.
%e A382659   Intersection with A147571 = {k : rad(k) = 210} is {210, 420, 630, 840, 1050, 1260, 1470} since m = 11^2 * 13 = 1573, etc.
%e A382659 For omega = 5, we have 2310 = 2*3*5*7*11, a term since 2310 < 13*17 = 2873; 2730 = 2*3*5*7*13 is not a term.
%e A382659 There are no terms larger than 2310, since the intersection with A147572 = {2310}, 2730 is not a term, and k = Product_{i=1..j} prime(i), k > prime(j+1)^2 * prime(j+2) for j > 5. Therefore the sequence is finite like A051250.
%t A382659 Select[Range[30030], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++]][[-1, 1]] ] ]
%Y A382659 Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A126706, A303554, A380539, A382248, A382960.
%K A382659 nonn,easy,fini,full
%O A382659 1,2
%A A382659 _Michael De Vlieger_, Apr 14 2025