OFFSET

1,1

COMMENTS

See A270971 for the motivation behind this sequence.

Riesel showed that there are infinitely many integers such that k*(2^m) - 1 is not prime for any integer m. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.

In this sequence, the lesser of (provable) Sierpiński pairs appears with the linear formula a(n) = 7523267 + 11184810*(n-1).

Since 7523267 is a term of A244561, for every integer k > 0, 7523267*2^k+1 has a divisor in the set {3, 5, 7, 13, 17, 241}. Because 11184810 = 2*3*5*7*13*17*241, a(n)*2^k+1 = 7523267*2^k+1 + 11184810*(n-1)*2^k+1 always has a divisor in the set {3, 5, 7, 13, 17, 241}. Since a(n) is always odd because of its definition, a(n) is a Sierpiński number. Additionally, 7523267 + 14 = 7523281 is also a term of A244561. So a(n) + 14 is a Sierpiński number too, with the same proof.

In conclusion, if the minimum difference between consecutive (provable) Sierpiński numbers is 14 (see comment section of A270971 for the reason behind this claim), a(n) and a(n) + 14 must be consecutive and a(n) = 7523267 + 11184810*(n-1) is the formula for this sequence.

EXAMPLE

7523267 is a term because 7523267 and 7523267 + 14 = 7523281 are consecutive (provable) Sierpiński numbers.

CROSSREFS

KEYWORD

nonn

AUTHOR

Altug Alkan, Mar 28 2016

STATUS

approved