A005529 Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found. (Formerly M1505)

2, 5, 17, 13, 37, 41, 101, 61, 29, 197, 113, 257, 181, 401, 97, 53, 577, 313, 677, 73, 157, 421, 109, 89, 613, 1297, 137, 761, 1601, 353, 149, 1013, 461, 1201, 1301, 541, 281, 2917, 3137, 673, 1741, 277, 1861, 769, 397, 241, 2113, 4357, 449, 2381, 2521, 5477

Offset

1,1

Comments

Primes associated with Stormer numbers.

See A002313 for the sorted list of primes. It can be shown that k^2 + 1 has at most one primitive prime factor; the other prime factors divide m^2 + 1 for some m < k. When k^2 + 1 has a primitive prime factor, k is a Stormer number (A005528), otherwise a non-Stormer number (A002312).

References

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Todd, Table of Arctangents. National Bureau of Standards, Washington, DC, 1951, p. vi.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Stormer Number.

Eric Weisstein's World of Mathematics, Primitive Prime Factor

Mathematica

prms={}; Do[f=First/@FactorInteger[k^2+1]; p=Complement[f, prms]; prms=Join[prms, p], {k, 100}]; prms

Prog

(Magma) V:=[]; for n in [1..75] do p:=Max([ x[1]: x in Factorization(n^2+1) ]); if not p in V then Append(~V, p); end if; end for; V; - Klaus Brockhaus, Oct 29 2008
(PARI) do(n)=my(v=List(), g=1, m, t, f); for(k=1, n, m=k^2+1; t=gcd(m, g); while(t>1, m/=t; t=gcd(m, t)); f=factor(m)[, 1]; if(#f, listput(v, f[1]); g*=f[1])); Vec(v) \\ Charles R Greathouse IV, Jun 11 2017

Crossrefs

Cf. A002312, A002313 (primes of the form 4k+1), A002522, A005528.

Sequence in context: A033835 A366609 A178198 * A259255 A274903 A206029

Adjacent sequences: A005526 A005527 A005528 * A005530 A005531 A005532

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

Edited by T. D. Noe, Oct 02 2003

Status

approved