This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330136 #12 Dec 15 2019 17:41:49
%S A330136 10,14,15,20,21,22,26,28,30,33,34,38,39,40,42,44,45,46,50,51,52,56,57,
%T A330136 58,60,62,63,66,68,69,70,74,75,76,78,80,82,84,86,87,88,90,92,93,94,98,
%U A330136 99,100,102,104,105,106,110,111,112,114,116,117,118,120,122,123
%N A330136 Numbers m such that 1 < gcd(m, 6) < m and m does not divide 6^e for e >= 0.
%C A330136 Numbers m that are neither 3-smooth nor reduced residues mod 6. Such numbers m have at least 1 prime factor p <= 3 and at least 1 prime factor q > 3.
%C A330136 Complement of the union of A003586 and A007310. Analogous to A105115 for A120944(2) = 10. This sequence applies to A120944(1) = 6 = A002110(1).
%C A330136 The only composite n in A024619 for which n < A096014(n) is 6. Let n be a composite that is not a prime power (i.e., in A024619), let p = lpf(n) = A020639(n), and let q = A053669(n) be the smallest prime that does not divide n. We observe that A096014(n) = A020639(n) * A053669(n) = pq. Such n with n < pq must minimize one factor while maximizing the other. The prime p is minimum when n is even, and q is greatest when n is the product p_k# of the smallest k primes, i.e., when n is in A002110. Alternatively, q is minimum when n is odd, however, n > 2p since n is the product of at least two distinct odd primes. Since p_k# greatly increases as k increments, while A053669(p_k#) = p_(k + 1), and observing that A096014(30) = 2 * 7 = 14, the only composite n in A024619 such that n < pq is 6.
%H A330136 Michael De Vlieger, <a href="/A330136/b330136.txt">Table of n, a(n) for n = 1..10000</a>
%e A330136 All m < 10 are not in the sequence since they either divide 6^e with integer e >= 0 or are coprime to 6.
%e A330136 10 is in the sequence since gcd(6, 10) = 2 and 10 does not divide 6^e with integer e >= 0.
%e A330136 11 is not in the sequence since 11 is coprime to 6.
%e A330136 12 is not in the sequence since 12 | 6^2.
%t A330136 With[{nn = 123, k = 6}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]
%Y A330136 Cf. A002110, A003586, A007310, A020639, A024619, A053669, A096014, A105115, A120944, A306999, A307589, A316991, A316992, A330137.
%K A330136 nonn,easy
%O A330136 1,1
%A A330136 _Michael De Vlieger_, Dec 02 2019