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%I #26 May 10 2020 08:22:26
%S 2,5,37,901,44101,5336101,901800901,260620460101,94083986096101,
%T 49770428644836901,41856930490307832901,40224510201185827416901,
%U 55067354465423397733736101,92568222856376731590410384101
%N a(n) = prime(n)#^2 + 1, where prime(n)# is the n-th primorial (A002110).
%C 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.
%C Comment from _Abhiram R Devesh_, Jan 23 2013: (Start)
%C (i) The last 3 digits of an entry is always either 101 or 901 (with the exception of the first 3 terms),
%C (ii) the thousand's place digit is an even number.
%C (End)
%H Vincenzo Librandi, <a href="/A189409/b189409.txt">Table of n, a(n) for n = 0..190</a>
%H E.W. Weisstein, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EuclidsTheorems.html">Euclid's Theorem</a>
%F a(n)=(E(n)-1)^2+1, where E(n) is the n-th Euclid number.
%e (p_16#)^2+1 = 1062053250251407755176413469419400772901 is prime.
%t Table[Product[Prime[n]^2, {n, 1, k}] + 1, {k, 0, 16}]
%t Join[{2},FoldList[Times,Prime[Range[20]]]^2+1] (* _Harvey P. Dale_, Jan 15 2019 *)
%o (Python)
%o from functools import reduce
%o import numpy as np
%o def factors(n):
%o return reduce(list.__add__, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
%o mul = 1
%o for i in range(1, 20):
%o if len(factors(i))<3:
%o mul *= i*i
%o print(mul+1, factors(mul+1))
%o # _Abhiram R Devesh_, Jan 23 2013
%o (PARI) list(maxx)={n=prime(1); cnt=0;print("0 2");
%o while(n<=maxx,q=(prodeuler(p=1,n,p))^2+1;cnt++;
%o print(cnt," ",q); n=nextprime(n+1)); } \\ _Bill McEachen_, Feb 03 2014
%Y A002110, A006862, A014545, A210482 (subsequence of primes).
%K nonn,easy
%O 0,1
%A _John M. Campbell_, Apr 21 2011
%E Typo in Mma fixed by _Vincenzo Librandi_, Feb 04 2014
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