login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 0
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 (list; graph; refs; listen; history; text; internal format)
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.

MAPLE

with(numtheory):

for n from 1 to 10000 do:

  x:=factorset(n^2+1):

   if issqrfree(n^2+1) = false

   then

    for m from 1 to n-1 do:

    y:=factorset(m^2+1):

       if x=y then printf (`%d %d \n`, n, m):

        else

        fi:

     od:

     fi:

    od:

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 28 12:46 EST 2022. Contains 350656 sequences. (Running on oeis4.)