A081545 Smallest prime which is 1 more than the product of n distinct composite numbers.
2, 5, 37, 193, 2161, 23041, 241921, 2903041, 55987201, 958003201, 17915904001, 250822656001, 5518098432001, 142216445952001, 2897001676800001, 90386452316160001, 1807729046323200001, 52563198423859200001
Offset: 0
Examples
Writing c(n) for the n-th composite number A002808(n):
a(0) = (Product_{i=1..0} c(i))+1 = 1+1 = 2 (empty product).
a(1) = c(1)+1 = 4+1 = 5.
a(2) = c(1)*c(4)+1 = 4*9+1 = 37, since c(1)*c(k)+1 is not prime for k < 4.
a(3) = c(1)*c(2)*c(3)+1 = 4*6*8 + 1 = 193.
a(4) = c(1)*c(2)*c(4)*c(5)+1 = 2161, nothing better since c(6)*c(3) > c(5)*c(4).
a(5) = c(1)*c(2)*c(3)*c(5)*c(6)+1 = 23041, none better since c(7)*c(4) > c(5)*c(6).
a(6) = c(1)*c(2)*c(3)*c(4)*c(5)*c(7)+1 = 241921, best since c(1)*...*c(6)+1 is not prime.
a(7) = p(7)+1 = 2903041 with p(n) = Product_{i=1..n} c(i). - _M. F. Hasler_, Jun 16 2007
Links
- Jinyuan Wang, Table of n, a(n) for n = 0..300 (terms 0..99 from M. F. Hasler)
Programs
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PARI
A081545(n, b=0 /*best*/, p=1 /*product*/, f=[]/*factors*/)={ if( #f
Formula
Theorem: For any m > 0 there is a K > 0 such that for all k > K, a(k)-1 is divisible by the first m composite numbers.
Corollary: For any b > 1, r > 0 there is a K > 0 such that for all k > K, a(k) == 1 (mod b^r). Taking b=10 shows that all a(k) > a(8) end in 0..01 with an increasing number of zeros. - M. F. Hasler, Jun 16 2007
Extensions
More terms from Michel ten Voorde Jun 13 2003
Terms beyond a(8) by M. F. Hasler Jun 16 2007
Comments