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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. 1
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

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Last modified August 15 07:14 EDT 2022. Contains 356128 sequences. (Running on oeis4.)