A291989 Smallest number that exceeds n and is divisible by at least one prime factor of n and by at least one prime that does not divide n.

6, 6, 6, 10, 10, 14, 10, 12, 12, 22, 14, 26, 18, 18, 18, 34, 20, 38, 22, 24, 24, 46, 26, 30, 28, 30, 30, 58, 33, 62, 34, 36, 36, 40, 38, 74, 40, 42, 42, 82, 44, 86, 46, 48, 48, 94, 50, 56, 52, 54, 54, 106, 56, 60, 58, 60, 60, 118, 62, 122, 66, 66, 66, 70, 68

Offset: 2

Author

Michael De Vlieger, Sep 20 2017

Keywords

• nonn
• easy

Comments

Numbers m in A096014 are even squarefree semiprimes, i.e., the product of A020639(n) and A053669(n). Numbers k in a(n) are always even composite, but not always squarefree or semiprime. For prime p, A096014(p) = a(p).

Let b(n) = A272619(n), continued for k > n that are products of at least one prime p that divides n and at least one prime q that is coprime to n. The index of a(n) in b(n) is A243823(n) + 1, i.e., a(n) is the term that would follow the terms of A272619(n), greater than n.

Examples

			a(6) = A096014(6) = 10 since for 6, among the next composites {8, 9, 10, ...}, 10 is the first that is divisible by at least one prime p = 2 | 6, and at least one prime 5 that is coprime to 6. Since A020639(6) = 2 and A053669(6) = 5, a(6) and A096014(6) are identical.
a(12) = 14 since 14 is both the next composite after 12, and divisible by at least one prime divisor 2 of 12 and one prime q = 7 that is coprime to 12. This differs from A096014(12) = 10 because A053669(12) = 5, and 2 * 5 = 10.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..10000
• Michael De Vlieger, Comparison of A096014 and A291989

Crossrefs

Cf. A096014, A100484, A243823, A272619.

Programs

Mathematica

Table[k = n + 2; While[Or[CoprimeQ[k, n], PowerMod[n, k, k] == 0], k++]; k, {n, 2, 66}] (* Michael De Vlieger, Sep 20 2017 *)

Formula

a(2) = 6; a(p) = A100484(pi(n)) for prime p > 2.

a(p) = A096014(p).