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A247339
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a(n) is the least number k such that the greatest prime divisor of k^2+1 is the smallest prime divisor of n^2+1.
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1
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1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 14, 1, 16, 1, 2, 1, 20, 1, 2, 1, 24, 1, 26, 1, 2, 1, 4, 1, 2, 1, 5, 1, 36, 1, 2, 1, 40, 1, 2, 1, 5, 1, 12, 1, 2, 1, 9, 1, 2, 1, 54, 1, 56, 1, 2, 1, 5, 1, 2, 1, 4, 1, 66, 1, 2, 1, 5, 1, 2, 1, 74, 1, 23, 1, 2, 1, 6, 1, 2
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OFFSET
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1,2
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COMMENTS
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a(n)=n if n^2+1 is prime and a(n)=1 if n is odd.
Conjecture: for all integer n, there exists at least an integer m <= n such that the smallest prime factor of n^2+1 is also the greatest prime factor of m^2+1. - Michel Lagneau, Sep 27 2015
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LINKS
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EXAMPLE
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a(34)=5 because the greatest prime divisor of 5^2+1 = 2*13 is the smallest prime divisor of 34^2+1 =13*89.
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MAPLE
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with(numtheory):nn:=2000:T:=array(1..nn):U:=array(1..nn):
for i from 1 to nn do:
x:=factorset(i^2+1):T[i]:=x[1]:U[i]:=i:
od:
for n from 1 to 100 do:
ii:=0:
for k from 1 to 50000 while(ii=0) do:
y:=factorset(k^2+1):n0:=nops(y):q:=y[n0]:
if q=T[n]
then
ii:=1: printf(`%d, `, k):
else
fi:
od:
od:
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MATHEMATICA
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Table[k = 1; While[FactorInteger[k^2 + 1][[-1, 1]] != FactorInteger[n^2 + 1][[1, 1]], k++]; k, {n, 82}] (* Michael De Vlieger, Sep 27 2015 *)
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PROG
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(PARI) a(n) = {f = factor(n^2+1)[1, 1]; k = 1; while (! ((g=factor(k^2+1)) && (g[#g~, 1] == f)), k++); k; } \\ Michel Marcus, Sep 14 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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