OFFSET
0,1
COMMENTS
A variation of Euclid numbers. It is unknown whether or not numbers in this sequence are always squarefree. It is unknown whether or not there exist infinitely many primes in this sequence. For Euclid numbers see A006862.
Comment from Abhiram R Devesh, Jan 23 2013: (Start)
(i) The last 3 digits of an entry is always either 101 or 901 (with the exception of the first 3 terms),
(ii) the thousand's place digit is an even number.
(End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..190
E.W. Weisstein, Integer Sequence Primes
Eric W. Weisstein's World of Mathematics, Euclid's Theorem
FORMULA
a(n)=(E(n)-1)^2+1, where E(n) is the n-th Euclid number.
EXAMPLE
(p_16#)^2+1 = 1062053250251407755176413469419400772901 is prime.
MATHEMATICA
Table[Product[Prime[n]^2, {n, 1, k}] + 1, {k, 0, 16}]
Join[{2}, FoldList[Times, Prime[Range[20]]]^2+1] (* Harvey P. Dale, Jan 15 2019 *)
PROG
(Python)
from functools import reduce
import numpy as np
def factors(n):
return reduce(list.__add__, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
mul = 1
for i in range(1, 20):
if len(factors(i))<3:
mul *= i*i
print(mul+1, factors(mul+1))
# Abhiram R Devesh, Jan 23 2013
(PARI) list(maxx)={n=prime(1); cnt=0; print("0 2");
while(n<=maxx, q=(prodeuler(p=1, n, p))^2+1; cnt++;
print(cnt, " ", q); n=nextprime(n+1)); } \\ Bill McEachen, Feb 03 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John M. Campbell, Apr 21 2011
EXTENSIONS
Typo in Mma fixed by Vincenzo Librandi, Feb 04 2014
STATUS
approved