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%I #15 Feb 21 2021 03:46:06
%S 1,6,10,12,8,12,10,16,20,18,16,20,14,24,32,16,36,20,24,40,28,20,40,36,
%T 22,32,32,30,28,48,26,48,60,40,40,32,54,56,40,44,32,48,34,60,80,64,42,
%U 40,52,50,72,40,80,44,84,48,64,108,44,60,80,46,64,56,72,96,52,68,50,88,96,70,84
%N a(1)=1; for n > 1, a(n) is the product of prime factors of the n-th composite, but with the largest prime incremented by 3 and the smallest decremented by 1.
%C In the prime number decomposition of k=A002808(n), one instance of the largest prime, pmax=A052369(n), is replaced by pmax+3 and one instance of the smallest prime, pmin=A056608(n), is replaced by pmin-1. a(n) is the product of this modified set of factors if nonprime. The case of n=1, k=4, is the only case where this modified product (2+3)*(2-1)=5 is prime and listed as a(1)=1.
%F a(n) = k*(pmax+3)*(pmin-1)/(pmin*pmax), n > 1, where k=A002808(n), pmin=A056608(n), pmax=A052369(n).
%e n-th composite
%e n & factorization transformed product
%e - --------------- --------------------------
%e 1 4 = 2*2 (2-1)*(2+3) = 1*5 = 5 (prime)
%e 2 6 = 2*3 (2-1)*(3+3) = 1*6 = 6 = a(2)
%e 3 8 = 2*2*2 (2-1)*2*(2+3) = 1*2*5 = 10 = a(3)
%e 4 9 = 3*3 (3-1)*(3+3) = 2*6 = 12 = a(4)
%e 5 10 = 2*5 (2-1)*(5+3) = 1*8 = 8 = a(5)
%e 6 12 = 2*2*3 (2-1)*2*(3+3) = 1*2*6 = 12 = a(6)
%e 7 14 = 2*7 (2-1)*(7+3) = 1*10 = 10 = a(7)
%K nonn
%O 1,2
%A _Juri-Stepan Gerasimov_, Aug 08 2008
%E Edited by _R. J. Mathar_, Aug 14 2008
%E Further edits by _Jon E. Schoenfield_, Feb 20 2021