A282092 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.

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

Offset

1,1

Comments

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), ...

Links

Table of n, a(n) for n=1..34.

Example

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.

Mathematica

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 *)

Prog

(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";
}
} # Daniel Suteu, Feb 08 2017
(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

Crossrefs

Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).

Cf. A002522, A059591, A059592, A124809.

Sequence in context: A203381 A207158 A304992 * A197938 A207153 A203222

Adjacent sequences: A282089 A282090 A282091 * A282093 A282094 A282095

Keyword

nonn

Author

Michel Lagneau, Feb 06 2017

Extensions

a(15)-a(29) from Daniel Suteu, Feb 08 2017

a(30) from Daniel Suteu, Feb 10 2017

a(31)-a(34) from Joerg Arndt, Feb 11 2017

Status

approved