OFFSET
1,1
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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MATHEMATICA
With[{nn = 123, k = 6}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Dec 02 2019
STATUS
approved