A246655 -id:A246655 - cp's OEIS frontend

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

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)
```

A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 12, 60, 60, 168, 504, 360, 660, 1092, 4080, 2448, 3420, 6072, 7800, 9828, 12180, 14880, 32736, 25308, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 262080, 150348, 178920, 194472, 246480, 265680, 285852, 352440, 456288, 515100, 546312, 612468, 647460

Offset: 1

Jianing Song, Apr 04 2022

nonn

For a communtative unital ring R, PSL_n(R), the projective special linear group of order n over R, is defined as SL_n(R)/{r*I_n: r^n = 1}. This is related to PGL_n(R), the projective general linear group of order n over R, which is defined as GL_n(R)/{r*I_n: r is a unit of R}.

Note that a(3) = a(4) = 60 refer to the same group (PSL(2,4) = PSL(2,5) = Alt(5)). Also PSL(2,9) = Alt(6).

			a(6) = 504 since A246655(6) = 8, so a(6) = 8*(8^2-1)/gcd(2,8-1) = 504.
a(7) = 360 since A246655(7) = 9, so a(7) = 9*(9^2-1)/gcd(2,9-1) = 360.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Projective special linear group

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): this sequence;
Aut(GL(2,q)): A353247;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.
A117762 is a subsequence, A335000 is a supersequence.

[(q+1)*q*(q-1)/gcd(2,q-1) | q <- [1..200], isprimepower(q)]

|PSL(2,q)| = q*(q^2-1)/2 if q is odd, q*(q^2-1) otherwise.
|PSL(2,q)| = |PGL(2,q)|/gcd(2,q-1) = |SL(2,q)|/gcd(2,q-1).
In general, |PSL(n,q)| = |PGL(n,q)|/gcd(n,q-1) = |SL(n,q)|/gcd(n,q-1).

A383294 Positions of prime powers (A246655) in EKG-sequence.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 14, 17, 20, 22, 24, 28, 31, 33, 37, 43, 50, 57, 61, 64, 67, 74, 76, 81, 89, 100, 107, 112, 115, 122, 124, 128, 134, 138, 151, 160, 167, 171, 182, 189, 197, 203, 207, 216, 232, 236, 240, 253, 259, 264, 279, 287, 290, 297, 305, 314, 319, 328, 336, 344, 359, 363, 371, 377, 381, 401, 420, 430, 438, 444

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence.

Positions of terms > 1 in A383293.
Disjoint union of A064955 and A383295.
Cf. A064413, A246655.

isA383294(n) = { my(x=A064413(n)); (isprime(x) || isprimepower(x)); };

A316190 a(n) is the distance between n and the nearest prime power (in the sense of A246655) other than n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3

Offset: 1

Peter Luschny, Jun 26 2018

nonn

			Note that 1369, 1373, 1381 and 1399 are prime powers. This leads to the mapping:
1373 -> 4,
1374 -> 1,
1375 -> 2,
1376 -> 3,
1377 -> 4,
1378 -> 3,
1379 -> 2,
1380 -> 1,
1381 -> 8.

S. M. Gonek, A formula of Landau and mean values of Zeta(s), Topics in Analytic Number Theory, ed. by S. W. Graham and J. D. Vaaler, 92-97, Univ. Texas Press 1985.
S. M. Gonek, An explicit formula of Landau and its applications to the theory of the zeta-function, Contemporary Math. 143 (1993), 395-413.

Peter Luschny, Table of n, a(n) for n = 1..10000
E. Landau, Über die Nullstellen der Zetafunktion, Math. Annalen 71, 548-564, (1911).

Cf. A246655, A316191.

A316190_list := proc(N) local a, b, d, m, k, P, R; R := NULL; m := 1;
P := select(t -> nops(numtheory:-factorset(t)) = 1 or t = 0, [$0..N]);
for k from 1 to nops(P)-1 do
    a := P[k]; b := P[k+1];
    if m = a then
       R := R, min(m - P[k-1] , b - m);
       m := m + 1;
    fi;
    while m < b do
        R := R, min(m - a , b - m);
        m := m + 1;
    od;
od; [R] end:
A316190_list(100);

a[n_] := Module[{k = 1}, While[!PrimePowerQ[n+k] && !PrimePowerQ[n-k], k++]; k]; Array[a, 100] (* Jean-François Alcover, Jul 25 2019 *)

A327297 Terms in A003656 that are not prime powers (A246655).

Original entry on oeis.org

12, 21, 24, 28, 33, 44, 56, 57, 69, 76, 77, 88, 92, 93, 124, 129, 133, 141, 152, 161, 172, 177, 184, 188, 201, 209, 213, 217, 236, 237, 248, 249, 253, 268, 284, 301, 309, 329, 332, 341, 344, 376, 381, 393, 412, 413, 417, 428, 437, 453, 472, 489, 497, 501, 508, 517, 524, 536, 537, 553, 556, 573, 581, 589, 597

Offset: 1

Jianing Song, Sep 16 2019

nonn

Conjecture: if D is a term of this sequence, then D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4. For example, a(1) = 12 = 3*4, a(2) = 21 = 3*7, a(3) = 24 = 3*8, a(4) = 28 = 4*7, a(5) = 33 = 3*11, ... [This conjecture is correct: see Theorem 1 and Theorem 2 of Ezra Brown link; see also A003656. - Jianing Song, Dec 28 2021]
Let k be the quadratic field with discriminant D, O_k be ring of integers of k, N(x) be the norm of x and (D/p) be the Kronecker symbol. If D is a term of this sequence and D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4, then:
(a) if (((-u)/p), ((-v)/p)) = (1, 1), (1, 0) or (0, 1), then N(x) = p has solutions in O_k, while N(y) = -p has no solutions in k. For example, for D = 21 and p = 37, we have ((-3)/37) = ((-7)/37) = 1, and N(x) = 37 has solution x = (13 + sqrt(21))/2, but N(y) = -37 has no solutions in Q(sqrt(21)).
(b) if (((-u)/p), ((-v)/p)) = (-1, -1), (-1, 0) or (0, -1), then N(x) = -p has solutions in O_k, while N(y) = p has no solutions in k. For example, for D = 12 and p = 11, we have ((-3)/11) = ((-4)/11) = -1, and N(x) = -11 has solution x = 1 + 2*sqrt(3), but N(y) = 11 has no solutions in Q(sqrt(3)).
(c) if (((-u)/p), ((-v)/p)) = (1, -1) or (-1, 1), then N(x) = +-p has no solutions in k.
The smallest number of the form above that is not in this sequence is 316 = 4*79.
Also, it is conjectured that the quadratic field with discriminant D has form class number 2, where D is a term of this sequence. This is equivalent to the conjecture above. [This can also be deduced from the first paragraph of Ezra Brown link: the norm of the fundamental unit of the field k is -1 if D = 8 or a prime congruent to 1 modulo 4, and 1 if D is in this sequence. Here k is the quadratic field with discriminant D. - Jianing Song, Dec 28 2021]

Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.

Subsequence of A003656 and A003658.
Complement of A003655 with respect to A003656.
Cf. A016105, A246655.

isA327297(D) = if(D>1&&isfundamental(D), quadclassunit(D)[1]==1&&!isprimepower(D), 0)

A327752 Primes powers (A246655) congruent to 1 mod 5.

Original entry on oeis.org

11, 16, 31, 41, 61, 71, 81, 101, 121, 131, 151, 181, 191, 211, 241, 251, 256, 271, 281, 311, 331, 361, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 841, 881, 911, 941, 961, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201

Offset: 1

Jianing Song, Sep 24 2019

nonn

Numbers k, not powers of 5, such that x^4 + x^3 + x^2 + x + 1 factors into four linear polynomials over GF(k).

This sequence consists of numbers of the form p^e where prime p == 1 (mod 5), p^(2e) where prime p == 4 (mod 5) and p^(4e) where prime p == 2, 3 (mod 5),

			k = 11: in GF(11), x^4 + x^3 + x^2 + x + 1 = (x - 3)*(x - 4)*(x - 5)*(x + 2);
k = 16: let GF(16) = GF(2)[y]/(y^4+y+1), then x^4 + x^3 + x^2 + x + 1 = (x - y^3)*(x - (y^3+y))*(x - (y^3+y^2))*(x - (y^3+y^2+y+1)).

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

Cf. A137827, A327753.

Intersection of A016861 and A246655.

[n:n in [2..1210]|IsPrimePower(n) and (n mod 5 eq 1)]; // Marius A. Burtea, Sep 26 2019

PARI
```
isok(n) = isprimepower(n) && (n%5==1)
```

A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

Original entry on oeis.org

2, 3, 5, 8, 17, 27, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 512, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1331, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663

Offset: 1

Martin Becker, May 18 2021

nonn

Also, prime powers q = p^(3^k) with prime p and nonnegative integer k and the property that q^2 + q + 1 is prime, since the exponent must be a power of 3, from the theory of cyclotomic polynomials. 17^(3^7) is in the sequence, generating a 5382-digit prime.

			5 = 5^1 is a term: 5^2 + 5 + 1 = 31 is prime.
8 = 2^3 is a term: 8^2 + 8 + 1 = 73 is prime.

Martin Becker, Table of n, a(n) for n = 1..20000

Intersection of A246655 and A002384.

Cf. A342691, A344448.

Select[Range@2000,PrimePowerQ@#&&PrimeQ[#^2+#+1]&] (* Giorgos Kalogeropoulos, May 18 2021 *)

N=50; i=0; a=vector(N); for(q=2, oo, if(isprimepower(q) && isprime(q^2+q+1), i+=1; a[i]=q; if(i==N, break))); a

A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 24, 120, 120, 336, 1512, 1440, 1320, 2184, 16320, 4896, 6840, 12144, 31200, 58968, 24360, 29760, 163680, 50616, 68880, 79464, 103776, 235200, 148824, 205320, 226920, 1572480, 300696, 357840, 388944, 492960, 2125440, 571704, 704880, 912576, 1030200, 1092624

Offset: 1

Jianing Song, Apr 04 2022

nonn

PGammaL_n(K) is the projective semilinear group of order n over K (see Wikipedia link). It is the semidirect product of PGL_n(K) and Aut(K), where Aut(K) is the group of field automorphisms of K. So if p is a prime, then PGammaL(n,p) is isomorphic to PGL(n,p).
We also have Aut(SL_n(K)) = Aut(PGL_n(K)) = Aut(PSL_n(K)) for arbitrary field K, and when n = 2 this is isomorphic to PGammaL_2(K). If n >= 3, this is isomorphic to the semidirect product of PGammaL_2(K) and C_2.
Examples are PGammaL(2,2) = S_3, PGammaL(2,3) = S_4, PGammaL(2,4) = PGammaL(2,5) = S_5, PGammaL(2,9) = Aut(S_6) = Aut(A_6).

			a(6) = 1512 since A246655(6) = 8 = 2^3, so a(6) = 3*A329119(6) = 3*504 = 1512.
a(7) = 1440 since A246655(7) = 9 = 3^2, so a(7) = 2*A329119(7) = 2*720 = 1440.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Projective semilinear group
Mathematics Stack Exchange, Do the groups SL, PGL, and PSL over a field K always have the same automorphism group?
Wikipedia, Semilinear map

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): A352806;
Aut(GL(2,q)): A353247;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): this sequence.

[(q+1)*q*(q-1)*isprimepower(q) | q <- [1..200], isprimepower(q)]

A353247 Orders of the finite groups Aut(GL_2(K)) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 48, 240, 480, 1344, 9072, 11520, 10560, 17472, 130560, 78336, 82080, 242880, 499200, 1415232, 584640, 476160, 4910400, 1214784, 2204160, 1907136, 4566144, 7526400, 7143552, 11497920, 7261440, 56609280, 12027840, 17176320, 18669312, 23662080, 136028160, 45736320, 56390400, 58404864, 82416000, 69927936

Offset: 1

Jianing Song, Apr 08 2022

nonn

For orders of Aut(SL_2(K)) = Aut(PGL_2(K)) = Aut(PSL_2(K)) see A352807.

See the Groupprops link for a formula for |Aut(GL(n,q))| in general.

			a(5) = 1344 since A246655(5) = 7, so a(5) = A352807(5)*eulerphi(2*(7-1)) = 336*4 = 1344.
a(6) = 9072 since A246655(6) = 8, so a(6) = A352807(6)*eulerphi(2*(8-1)) = 1512*6 = 9072.
a(7) = 11520 since A246655(7) = 9, so a(7) = A352807(7)*eulerphi(2*(9-1)) = 1440*8 = 15120.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Endomorphism structure of general linear group over a finite field

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): A352806;
Aut(GL(2,q)): this sequence;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.

[(q+1)*q*(q-1)*isprimepower(q)*eulerphi(2*(q-1)) | q <- [1..200], isprimepower(q)]

A365082 Prime powers (A246655) q such that -2 is a nonzero square in the finite field F_q.

Original entry on oeis.org

3, 9, 11, 17, 19, 25, 27, 41, 43, 49, 59, 67, 73, 81, 83, 89, 97, 107, 113, 121, 131, 137, 139, 163, 169, 179, 193, 211, 227, 233, 241, 243, 251, 257, 281, 283, 289, 307, 313, 331, 337, 347, 353, 361, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 529

Offset: 1

Jianing Song, Oct 22 2023

Prime powers q that are congruent to 1 or 3 modulo 8 (see A366526).
Odd prime powers q such that (-2)^((q-1)/2) = 1 in F_q.
Prime powers q such that x^2 + 2 splits into different linear factors in F_q[x].
Contains the powers of primes congruent to 1 or 3 modulo 8 and the even powers of primes congruent to 5 or 7 modulo 8.

			49 is a term since -2 = -9 = (+-3i)^2 in F_49 = F_7(i).

Jianing Song, Table of n, a(n) for n = 1..10000

Supersequence of A033200.

Prime powers q such that a is a nonzero square in F_q: this sequence (q=-2), A085759 (q=-1), A366526 (q=2), A365313 (q=3).

isA365082(n) = isprimepower(n) && (n%8==1 || n%8==3)

cp's OEIS Frontend

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

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Magma

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A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

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A383294 Positions of prime powers (A246655) in EKG-sequence.

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A316190 a(n) is the distance between n and the nearest prime power (in the sense of A246655) other than n.

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A327297 Terms in A003656 that are not prime powers (A246655).

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A327752 Primes powers (A246655) congruent to 1 mod 5.

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Magma

PARI

A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

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Programs

Mathematica

PARI

A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

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