[seqfan] On A035095

Vladimir Shevelev shevelev at bgu.ac.il
Mon Dec 1 16:42:31 CET 2014


Dear Seq Fans,

A035095 is a classic sequence which is "a version of the
"least prime in special arithmetic progressions" problem."[N. Sloane]
A035095(n) is the smallest prime congruent to 1 mod n-th prime.
I noted that it is also minimum of the smallest prime factors of 
F(n,i) = (i^prime(n)-1)/(i-1), when i runs all integers in 
[2, prime(n)]. Every prime factor of F(n,i) is congruent
to 1 modulo prime(n). Indeed, for every considered i,
F(n,i) is either prime or overpseudoprime to base i (see
https://cs.uwaterloo.ca/journals/JIS/VOL15/Castillo/castillo2.pdf,
Theorem 25).
By handy I found for the first five n the suitable minimal 
values of i=i(n) on which the smallest prime factor of F(n,i(n))
is A035095(n). Peter Moses extended the sequence {i(n)} up to n=16:
2,2,3,7,2,10,8,5,2,3,7,5,3,6,4,3,        (*)
but he wrote me that "so far, the computer has been working
on n=17 for over 16 hours!"
I think that sequence (*) is very interesting and useful  for a further research.
 If anyone could propose a simpler algorithm to calculate numbers (*)?

Best regards,
Vladimir


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