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

0,1

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

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

Sequence in context: A352973 A041959 A051739 * A222318 A084436 A210482

Adjacent sequences: A189406 A189407 A189408 * A189410 A189411 A189412

Keyword

nonn,easy

Author

John M. Campbell, Apr 21 2011

Extensions

Typo in Mma fixed by Vincenzo Librandi, Feb 04 2014

Status

approved