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

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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

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

Programs

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

Mathematica

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

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

Extensions

Edited by T. D. Noe, Oct 02 2003