A189409 a(n) = prime(n)#^2 + 1, where prime(n)# is the n-th primorial (A002110).

2, 5, 37, 901, 44101, 5336101, 901800901, 260620460101, 94083986096101, 49770428644836901, 41856930490307832901, 40224510201185827416901, 55067354465423397733736101, 92568222856376731590410384101

Offset: 0

Author

John M. Campbell, Apr 21 2011

Keywords

• nonn
• easy

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)

Examples

			(p_16#)^2+1 = 1062053250251407755176413469419400772901 is prime.
		

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

Crossrefs

A002110, A006862, A014545, A210482 (subsequence of primes).

Programs

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 *)

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

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

Formula

a(n)=(E(n)-1)^2+1, where E(n) is the n-th Euclid number.

Extensions

Typo in Mma fixed by Vincenzo Librandi, Feb 04 2014