A330136 Numbers m such that 1 < gcd(m, 6) < m and m does not divide 6^e for e >= 0.

10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 45, 46, 50, 51, 52, 56, 57, 58, 60, 62, 63, 66, 68, 69, 70, 74, 75, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123

Offset: 1

Author

Michael De Vlieger, Dec 02 2019

Keywords

• nonn
• easy

Comments

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.

Complement of the union of A003586 and A007310. Analogous to A105115 for A120944(2) = 10. This sequence applies to A120944(1) = 6 = A002110(1).

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.

Examples

			All m < 10 are not in the sequence since they either divide 6^e with integer e >= 0 or are coprime to 6.
10 is in the sequence since gcd(6, 10) = 2 and 10 does not divide 6^e with integer e >= 0.
11 is not in the sequence since 11 is coprime to 6.
12 is not in the sequence since 12 | 6^2.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002110, A003586, A007310, A020639, A024619, A053669, A096014, A105115, A120944, A306999, A307589, A316991, A316992, A330137.

Programs

Mathematica

With[{nn = 123, k = 6}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]