A125258 -id:A125258 - cp's OEIS frontend

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

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

Offset: 1

Eric Chen, Nov 13 2014

nonn

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

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

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.

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

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

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

A125256 Smallest odd prime divisor of n^2 + 1.

Original entry on oeis.org

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

Offset: 2

Nick Hobson, Nov 26 2006

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

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

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

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

Cf. A002522, A002496, A014442, A057207, A031439.
For bisections see A256970, A293958.
See also A125257, A125258, A294656, A294657, A294658.

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

Table[Select[First/@FactorInteger[n^2+1],OddQ][[1]],{n,2,68}] (* James C. McMahon, Dec 16 2024 *)

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

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

cp's OEIS Frontend

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

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A125256 Smallest odd prime divisor of n^2 + 1.

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Links

Crossrefs

Programs

Maple

Mathematica

PARI

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