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%I #10 Sep 26 2017 08:27:26
%S 6,6,6,10,10,14,10,12,12,22,14,26,18,18,18,34,20,38,22,24,24,46,26,30,
%T 28,30,30,58,33,62,34,36,36,40,38,74,40,42,42,82,44,86,46,48,48,94,50,
%U 56,52,54,54,106,56,60,58,60,60,118,62,122,66,66,66,70,68
%N 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.
%C 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).
%C 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.
%H Michael De Vlieger, <a href="/A291989/b291989.txt">Table of n, a(n) for n = 2..10000</a>
%H Michael De Vlieger, <a href="/A291989/a291989.txt">Comparison of A096014 and A291989</a>
%F a(2) = 6; a(p) = A100484(pi(n)) for prime p > 2.
%F a(p) = A096014(p).
%e 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.
%e 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.
%t 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 *)
%Y Cf. A096014, A100484, A243823, A272619.
%K nonn,easy
%O 2,1
%A _Michael De Vlieger_, Sep 20 2017
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