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

%I A189409 #27 Feb 16 2025 08:33:14
%S A189409 2,5,37,901,44101,5336101,901800901,260620460101,94083986096101,
%T A189409 49770428644836901,41856930490307832901,40224510201185827416901,
%U A189409 55067354465423397733736101,92568222856376731590410384101
%N A189409 a(n) = prime(n)#^2 + 1, where prime(n)# is the n-th primorial (A002110).
%C A189409 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 A189409 Comment from _Abhiram R Devesh_, Jan 23 2013: (Start)
%C A189409 (i) The last 3 digits of an entry is always either 101 or 901 (with the exception of the first 3 terms),
%C A189409 (ii) the thousand's place digit is an even number.
%C A189409 (End)
%H A189409 Vincenzo Librandi, <a href="/A189409/b189409.txt">Table of n, a(n) for n = 0..190</a>
%H A189409 E.W. Weisstein, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A189409 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EuclidsTheorems.html">Euclid's Theorem</a>
%F A189409 a(n)=(E(n)-1)^2+1, where E(n) is the n-th Euclid number.
%e A189409 (p_16#)^2+1 = 1062053250251407755176413469419400772901 is prime.
%t A189409 Table[Product[Prime[n]^2, {n, 1, k}] + 1, {k, 0, 16}]
%t A189409 Join[{2},FoldList[Times,Prime[Range[20]]]^2+1] (* _Harvey P. Dale_, Jan 15 2019 *)
%o A189409 (Python)
%o A189409 from functools import reduce
%o A189409 import numpy as np
%o A189409 def factors(n):
%o A189409     return reduce(list.__add__, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
%o A189409 mul = 1
%o A189409 for i in range(1, 20):
%o A189409     if len(factors(i))<3:
%o A189409         mul *= i*i
%o A189409         print(mul+1, factors(mul+1))
%o A189409 # _Abhiram R Devesh_, Jan 23 2013
%o A189409 (PARI) list(maxx)={n=prime(1); cnt=0;print("0  2");
%o A189409 while(n<=maxx,q=(prodeuler(p=1,n,p))^2+1;cnt++;
%o A189409 print(cnt,"  ",q); n=nextprime(n+1)); } \\ _Bill McEachen_, Feb 03 2014
%Y A189409 A002110, A006862, A014545, A210482 (subsequence of primes).
%K A189409 nonn,easy
%O A189409 0,1
%A A189409 _John M. Campbell_, Apr 21 2011
%E A189409 Typo in Mma fixed by _Vincenzo Librandi_, Feb 04 2014