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A189409
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a(n) = prime(n)#^2 + 1, where prime(n)# is the n-th primorial (A002110).
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3
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2, 5, 37, 901, 44101, 5336101, 901800901, 260620460101, 94083986096101, 49770428644836901, 41856930490307832901, 40224510201185827416901, 55067354465423397733736101, 92568222856376731590410384101
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listen;
history;
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internal format)
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OFFSET
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0,1
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COMMENTS
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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.
(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)
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LINKS
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FORMULA
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a(n)=(E(n)-1)^2+1, where E(n) is the n-th Euclid number.
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EXAMPLE
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(p_16#)^2+1 = 1062053250251407755176413469419400772901 is prime.
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MATHEMATICA
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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 *)
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PROG
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(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))
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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