A124990 -id:A124990

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A246397

Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

+10
8

2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers n such that n^4-n^2+1 is prime, or numbers n such that A060886(n) is prime.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

MAPLE

A246397:=n->`if`(isprime(n^4-n^2+1), n, NULL): seq(A246397(n), n=1..300); # Wesley Ivan Hurt, Nov 14 2014

MATHEMATICA

Select[Range[350], PrimeQ[Cyclotomic[12, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

PROG

(PARI) for(n=1, 10^3, if(isprime(polcyclo(12, n)), print1(n, ", "))); \\ Joerg Arndt, Nov 13 2014

CROSSREFS

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862 (11), this sequence (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060886, A125258, A085398, A124990.

KEYWORD

nonn

AUTHOR

Eric Chen, Nov 13 2014

STATUS

approved

A125258

Smallest prime divisor of n^4-n^2+1.

+10
3

13, 73, 241, 601, 13, 13, 37, 6481, 9901, 13, 20593, 28393, 37, 13, 97, 83233, 229, 13, 13, 61, 157, 37, 13, 390001, 181, 530713, 13, 37, 809101, 922561, 13, 13, 1069, 277, 1678321, 13, 2083693, 2311921, 61, 13, 673, 3416953, 1753, 13, 13, 1213, 5306113

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

All divisors of n^4-n^2+1 are congruent to 1 modulo 12.

a(n) = 13 if and only if n is congruent to 2, -2, 6, or -6 modulo 13.

REFERENCES

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, NY, Second Edition (1990), p. 63.

LINKS

Nick Hobson, Table of n, a(n) for n = 2..1000

EXAMPLE

The prime divisors of 6^4-6^2+1=1261 are 13 and 97, so a(5) = 13.

MATHEMATICA

Table[FactorInteger[n^4-n^2+1][[1, 1]], {n, 2, 50}] (* Harvey P. Dale, Feb 27 2012 *)

PROG

(PARI) vector(49, n, if(n<2, "-", factor(n^4-n^2+1)[1, 1]))

CROSSREFS

Cf. A060886, A124990.

KEYWORD

easy,nonn

AUTHOR

Nick Hobson, Nov 26 2006

STATUS

approved

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