A141467 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.

1, 6, 10, 12, 8, 12, 10, 16, 20, 18, 16, 20, 14, 24, 32, 16, 36, 20, 24, 40, 28, 20, 40, 36, 22, 32, 32, 30, 28, 48, 26, 48, 60, 40, 40, 32, 54, 56, 40, 44, 32, 48, 34, 60, 80, 64, 42, 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

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 08 2008

Keywords

• nonn

Comments

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.

Examples

			     n-th composite
  n  & factorization      transformed product
  -  ---------------  --------------------------
  1      4 = 2*2      (2-1)*(2+3)   = 1*5   =  5 (prime)
  2      6 = 2*3      (2-1)*(3+3)   = 1*6   =  6 = a(2)
  3      8 = 2*2*2    (2-1)*2*(2+3) = 1*2*5 = 10 = a(3)
  4      9 = 3*3      (3-1)*(3+3)   = 2*6   = 12 = a(4)
  5     10 = 2*5      (2-1)*(5+3)   = 1*8   =  8 = a(5)
  6     12 = 2*2*3    (2-1)*2*(3+3) = 1*2*6 = 12 = a(6)
  7     14 = 2*7      (2-1)*(7+3)   = 1*10  = 10 = a(7)
		

Formula

a(n) = k*(pmax+3)*(pmin-1)/(pmin*pmax), n > 1, where k=A002808(n), pmin=A056608(n), pmax=A052369(n).

Extensions

Edited by R. J. Mathar, Aug 14 2008

Further edits by Jon E. Schoenfield, Feb 20 2021