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URL: https://oeis.org/A078598

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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.
0
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.
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
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