A125258 -id:A125258

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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

A125256

Smallest odd prime divisor of n^2 + 1.

+10
7

5, 5, 17, 13, 37, 5, 5, 41, 101, 61, 5, 5, 197, 113, 257, 5, 5, 181, 401, 13, 5, 5, 577, 313, 677, 5, 5, 421, 17, 13, 5, 5, 13, 613, 1297, 5, 5, 761, 1601, 29, 5, 5, 13, 1013, 29, 5, 5, 1201, 41, 1301, 5, 5, 2917, 17, 3137, 5, 5, 1741, 13, 1861, 5, 5, 17, 2113, 4357, 5, 5

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

OFFSET

2,1

COMMENTS

Any odd prime divisor of n^2+1 is congruent to 1 modulo 4.

n^2+1 is never a power of 2 for n > 1; hence a prime divisor congruent to 1 modulo 4 always exists.

a(n) = 5 if and only if n is congruent to 2 or -2 modulo 5.

If the map "x -> smallest odd prime divisor of n^2+1" is iterated, does it always terminate in the 2-cycle (5 <-> 13)? - Zoran Sunic, Oct 25 2017

REFERENCES

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

LINKS

Ray Chandler, Table of n, a(n) for n = 2..20001 (first 999 terms from Nick Hobson)

EXAMPLE

The prime divisors of 8^2 + 1 = 65 are 5 and 13, so a(7) = 5.

MAPLE

with(numtheory, factorset);

A125256 := proc(n) local t1, t2;

if n <= 1 then return(-1); fi;

if (n mod 5) = 2 or (n mod 5) = 3 then return(5); fi;

t1 := numtheory[factorset](n^2+1);

t2:=sort(convert(t1, list));

if (n mod 2) = 1 then return(t2[2]); fi;

t2[1];

end;

[seq(A125256(n), n=1..40)]; # N. J. A. Sloane, Nov 04 2017

PROG

(PARI) vector(68, n, if(n<2, "-", factor(n^2+1)[1+(n%2), 1]))

(PARI) A125256(n)=factor(n^2+1)[1+bittest(n, 0), 1] \\ M. F. Hasler, Nov 06 2017

CROSSREFS

Cf. A002522, A002496, A014442, A057207, A031439.

For bisections see A256970, A293958.

See also A125257, A125258, A294656, A294657, A294658.

KEYWORD

easy,nonn

AUTHOR

Nick Hobson, Nov 26 2006

STATUS

approved

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