

A065876


a(n) is the smallest m > n such that n^2 + 1 divides m^2 + 1.


7



1, 3, 3, 7, 13, 21, 31, 43, 18, 73, 91, 111, 17, 47, 183, 211, 241, 133, 57, 343, 381, 47, 172, 83, 553, 601, 651, 173, 342, 813, 242, 265, 132, 403, 411, 1191, 1261, 237, 327, 1483, 1561, 1641, 748, 857, 850, 1981, 684, 463, 413, 2353, 255, 2551, 593, 1177, 2863, 123, 3081, 307, 1288, 3423
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OFFSET

0,2


COMMENTS

a(n) exists because n^2 + 1 divides (n^2  n + 1)^2 + 1. The set of n such a(n) = n^2  n + 1 is S = (2, 3, 4, 5, 6, 7, 9, 11, 14, 15, ...).
a(n) = n^2  n + 1 whenever n^2 + 1 is prime or twice a prime. Up to n=1000, the only other n for which a(n) = n^2  n + 1 are 7, 41 and 239. Is it a coincidence that these are NSW primes (A088165)?  Franklin T. AdamsWatters, Oct 17 2006
It appears that the density of even numbers in this sequence approaches a limit near 1/4. It appears that the density of even values for indices where a(n) != n^2  n + 1 is approaching a number near 1/4 and based on the previous comment the density of n for which a(n) = n^2  n + 1 is almost certainly 0.  Franklin T. AdamsWatters, Oct 17 2006


LINKS

Franklin T. AdamsWatters, Table of n, a(n) for n = 0..1000


MATHEMATICA

Do[k = 1; While[m = (k^2 + 1)/(n^2 + 1); m < 2  !IntegerQ[m], k++ ]; Print[k], {n, 1, 40 } ]


PROG

(PARI) { for (n=0, 1000, a=n+1; while ((a^2 + 1)%(n^2 + 1) != 0, a++); write("b065876.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 03 2009


CROSSREFS

Cf. A002731, A005574, A071557, A088165.
Sequence in context: A096188 A187873 A306665 * A204858 A095008 A214825
Adjacent sequences: A065873 A065874 A065875 * A065877 A065878 A065879


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Dec 07 2001


EXTENSIONS

More terms from Robert G. Wilson v, Dec 11 2001
Further terms from Franklin T. AdamsWatters, Oct 17 2006


STATUS

approved



