A078598 Primes p for which one of the two factors in the factorization of the polynomial Sum_{i=0..p-1} p^i*x^(2i) has only positive coefficients.

5, 13, 29, 37, 61, 101, 109, 157, 181, 229, 277, 349, 373, 421, 541, 613, 661, 709, 757, 829, 1021, 1069, 1093, 1117, 1213, 1381, 1429, 1453, 1549, 1621, 1669, 1741, 1789, 1933, 2029, 2053, 2221

Offset

1,1

Comments

This polynomial factors iff p == 1 mod 4, but the positivity behavior seems restricted to p == 5 mod 8 with 149 as the first exception.

Links

Table of n, a(n) for n=1..37.

Example

For p=5 we have 625x^8+125x^6+25x^4+5x^2+1=(25x^4+25x^3+15x^2+5x+1)(25x^4-25x^3+15x^2-5x+1)

Prog

(PARI) posipoly(pol) = {for (i=1, poldegree(pol), if (polcoeff(pol, i) <0, return(0)); ); return (1); }
lista(n) = {forprime(p=3, n, fp = factor(sum(i=0, p-1, p^i*x^(2*i))); if (matsize(fp) == [2, 2], if (posipoly(fp[2, 1]) || posipoly(fp[1, 1]), print1(p, ", ")); ); ); } \\ Michel Marcus, Apr 14 2013

Crossrefs

Sequence in context: A213050 A216822 A217466 * A155054 A158756 A185086

Adjacent sequences: A078595 A078596 A078597 * A078599 A078600 A078601

Keyword

nonn

Author

Aart Blokhuis (aartb(AT)win.tue.nl), Dec 09 2002

Extensions

Corrected and extended by Michel Marcus, Apr 14 2013

Added more terms, Joerg Arndt, Apr 19 2013

Status

approved