A291989 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

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.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Michael De Vlieger, Comparison of A096014 and A291989

FORMULA

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

a(p) = A096014(p).

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A096014, A100484, A243823, A272619.

Sequence in context: A179409 A186983 A046264 * A035019 A216057 A212096

Adjacent sequences: A291986 A291987 A291988 * A291990 A291991 A291992

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Sep 20 2017

STATUS

approved