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A282092
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Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.
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1
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7, 18, 117, 239, 378, 843, 2207, 2943, 4443, 4662, 6072, 8307, 8708, 9872, 31561, 103682, 271443, 853932, 1021693, 3539232, 3699356, 6349657, 6907607, 7042807, 7249325, 9335094, 12623932, 12752043, 12813848, 22211431, 33385282, 42483057, 52374157, 105026693
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OFFSET
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1,1
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COMMENTS
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For the pairs (m, k), is k always unique?
The pairs (m, k) are (7, 3), (18, 8), (117, 43), (239, 5), (378, 132), (843, 377), (2207, 987), (2943, 73), (4443, 53), (4662, 1568), (6072, 5118), (8307, 743), (8708, 2112), (9872, 2738), ...
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LINKS
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EXAMPLE
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7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5.
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MATHEMATICA
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Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* Michael De Vlieger, Feb 07 2017 *)
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PROG
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(Perl)
use ntheory qw(:all);
for (my ($m, %t) = 1 ; ; ++$m) {
my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
push @{$t{$k}}, $m;
if (@{$t{$k}} >= 2) {
print'('.join(', ', reverse(@{$t{$k}})).")\n";
}
(PARI) isok(n)=ok = 0; vn = factor(n^2+1)[, 1]; for (k=1, n-1, if (factor(k^2+1)[, 1] == vn, ok = 1; break); ); ok; \\ Michel Marcus, Feb 09 2017
(PARI) squeeze(f)=factorback(f)\2
list(lim)=my(v=List(), m=Map(), t); for(n=1, lim, t=squeeze(factor(n^2+1)[, 1]); if(mapisdefined(m, t), listput(v, n), mapput(m, t, 0))); Vec(v) \\ Charles R Greathouse IV, Feb 12 2017
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CROSSREFS
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Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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