A327752 -id:A327752 - cp's OEIS frontend

A137827 Prime powers (A246655) congruent to 1 (mod 3).

4, 7, 13, 16, 19, 25, 31, 37, 43, 49, 61, 64, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151, 157, 163, 169, 181, 193, 199, 211, 223, 229, 241, 256, 271, 277, 283, 289, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499

Offset: 1

Eric W. Weisstein, Feb 12 2008

Numbers k not being powers of 3 such that x^2 + x + 1 (or x^2 - x + 1) is reducible over GF(k). - Jianing Song, Sep 24 2019

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cyclotomic Graph

Row 3 of A354940 (conjectured).
Intersection of A016777 and A246655.
Cf. A354982 (characteristic function), A354984 (terms * 3).
Cf. also A327752, A327753.

Select[ Range[4, 500], (Mod[#, 3] == 1 && Mod[#, # - EulerPhi[#]] == 0)& ] (* Jean-François Alcover, Oct 26 2012 *)

isok(n) = isprimepower(n) && ((n % 3) == 1); \\ Michel Marcus, Sep 24 2019

A327753 Primes powers (A246655) congruent to 4 mod 5.

Original entry on oeis.org

4, 9, 19, 29, 49, 59, 64, 79, 89, 109, 139, 149, 169, 179, 199, 229, 239, 269, 289, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 529, 569, 599, 619, 659, 709, 719, 729, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1024, 1039, 1049, 1069, 1109, 1129, 1229, 1249

Offset: 1

Jianing Song, Sep 24 2019

nonn

Numbers k such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(k).
Note that x^4 + x^3 + x^2 + x + 1 is reducible over GF(k) if and only if there exists some a in GF(k) such that a^2 - a - 1 = 0, and then x^4 + x^3 + x^2 + x + 1 = (x^2 + a*x + 1) * (x^2 + (1-a)*x + 1). There exists some a in GF(k) such that a^2 - a - 1 = 0 if and only if kronecker(k,5) = 1, or k == 1, 4 (mod 5). If k == 1 (mod 5), then x^4 + x^3 + x^2 + x + 1 can be further factored into four linear polynomials.
This sequence consists of numbers of the form p^(2e+1) where prime p == 4 (mod 5) and p^(4e+2) where prime p == 2, 3 (mod 5),

			k = 4: let GF(4) = GF(2)[w], w^2 + w + 1 = 0, then x^4 + x^3 + x^2 + x + 1 = (x^2 + w*x + 1)*(x^2 + (w+1)*x + 1);
k = 9: let GF(9) = GF(3)[i], i^2 = -1, then x^4 + x^3 + x^2 + x + 1 = (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
k = 19: in GF(19), x^4 + x^3 + x^2 + x + 1 = (x^2 + 5x + 1)*(x^2 - 4x + 1).

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A137827, A327752.

Intersection of A016897 and A246655.

[n:n in [2..1250]|IsPrimePower(n) and (n mod 5 eq 4)]; // Marius A. Burtea, Sep 26 2019

Select[Range@ 1250, And[PrimePowerQ@ #, Mod[#, 5] == 4] &] (* Michael De Vlieger, Sep 27 2019 *)

PARI
```
isok(n) = isprimepower(n) && (n%5==4)
```

cp's OEIS Frontend

A137827 Prime powers (A246655) congruent to 1 (mod 3).

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