A033200 -id:A033200 - cp's OEIS frontend

A296938 Rational primes that decompose in the field Q(sqrt(17)).

2, 13, 19, 43, 47, 53, 59, 67, 83, 89, 101, 103, 127, 137, 149, 151, 157, 179, 191, 223, 229, 239, 251, 257, 263, 271, 281, 293, 307, 331, 349, 353, 359, 373, 383, 389, 409, 421, 433, 443, 457, 461, 463, 467, 491, 509, 523, 557, 563, 569, 577, 587, 593, 599

Offset: 1

N. J. A. Sloane, Dec 26 2017

From Jianing Song, Apr 21 2022: (Start)
Primes p such that kronecker(17, p) = kronecker(p, 17) = 1, where kronecker() is the kronecker symbol. That is to say, primes p that are quadratic residues modulo 17.
Primes p such that p^8 == 1 (mod 17).
Primes p == 1, 2, 4, 8, 9, 13, 15, 16 (mod 17). (End)

Cf. A011584 (kronecker symbol modulo 17).
Rational primes that decompose in the quadratic field with discriminant D: A139513 (D=-20), A191019 (D=-19), A191018 (D=-15), A296920 (D=-11), A033200 (D=-8), A045386 (D=-7), A002144 (D=-4), A002476 (D=-3), A045468 (D=5), A001132 (D=8), A097933 (D=12), A296937 (D=13), this sequence (D=17).
Cf. A038890 (inert rational primes in the field Q(sqrt(17))).

[p: p in PrimesUpTo(600) | KroneckerSymbol(p, 17) eq 1]; // Vincenzo Librandi, Apr 09 2020

Load the Maple program HH given in A296920. Then run HH(17, 200); This produces A296938, A038890, A038889.

Select[Prime[Range[200]], JacobiSymbol[#, 17]==1&] (* Vincenzo Librandi, Apr 09 2020 *)

isA296938(p) = isprime(p) && kronecker(p,17) == 1 \\ Jianing Song, Apr 21 2022

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)

A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

Original entry on oeis.org

0, 1, 3, -6, -90, 45, 5607, 8316, -616572, -2517075, 106354215, 779869134, -26562900078, -299503403199, 9075456298755, 144911485323000, -4066415773786872, -87372799002303111, 2313066895842715947, 64609858869087786210, -1627745411473223627970

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 or 3 (mod 8), equivalently, rational primes that split in the field extension Q(sqrt(-2)) of Q. See A033200.
Moll's conjecture 5.5 extends to this sequence: for primes p of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 5 or 7 (mod 8), equivalently, primes that remain inert in the field extension Q(sqrt(-2)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-2)). See A033203.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(n) : n = 1..100] =  [0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 9, 9, 9, 10, 10, 10, 11, 11, 11, 13, 13, 13, 14, 14, 14, 15, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 19, 22, 22, 22, 23, 23, 23, 24, 24, 24, 26, 26, 26, 27, 27, 27, 28, 28, 28, 30, 30, 30, 31, 31, 31, 32, 32, 32, 36, 36, 36, 37, 37, 37, 38, 38, 38, 41, 40, 40, 42, 41, 41, 42, 42, 42, 44, 44, 44, 45, 45, 45, 46, 46, 46, 49, 49, 49, 50].
Note that v_3(a(100)) = 50 = 100/(3 - 1), in agrement with the asymptotic growth for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(n) : n = 1..100] = [0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else ( (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) with a(0) = 0 and a(1) = 1.

a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1,n+1-2*k).

A374294 a(n) is the smallest positive integer k such that A002325(k) = n.

Original entry on oeis.org

1, 3, 9, 27, 81, 99, 729, 297, 1089, 891, 59049, 1683, 531441, 8019, 9801, 5049, 43046721, 18513, 387420489, 15147, 88209, 649539, 31381059609, 31977, 1185921, 5845851, 314721, 136323, 22876792454961, 166617, 205891132094649, 95931, 7144929, 473513931, 10673289, 351747, 150094635296999121

Offset: 1

Seiichi Manyama, Jul 02 2024

nonn

			   n |  a(n)
-----+-----------------------
   2 |     3.
   3 |     9 = 3^2.
   4 |    27 = 3^3.
   5 |    81 = 3^4.
   6 |    99 = 3^2 * 11.
   7 |   729 = 3^6.
   8 |   297 = 3^3 * 11.
   9 |  1089 = 3^2 * 11^2.
  10 |   891 = 3^4 * 11.
  11 | 59049 = 3^10.
  12 |  1683 = 3^2 * 11 * 17.

Cf. A002325, A033715, A033200, A200977, A374295.

If p is prime, a(p) = 3^(p-1).

a(n) is divisible by 3 for n > 1.

A201544 Odd numbers of the form x^2 + 2*y^2 with positive integers x and y.

Original entry on oeis.org

3, 9, 11, 17, 19, 27, 33, 41, 43, 51, 57, 59, 67, 73, 75, 81, 83, 89, 97, 99, 107, 113, 121, 123, 129, 131, 137, 139, 147, 153, 163, 171, 177, 179, 187, 193, 201, 209, 211, 219, 225, 227, 233, 241, 243, 249, 251, 257, 267, 275, 281, 283, 289, 291, 297, 307

Offset: 1

Zak Seidov, Dec 02 2011

nonn

All terms == {1,3} mod 8. Terms that are not multiple of some previous term are prime numbers (see A033203, except for the first term 2 there).

For the numbers with positive proper representations see A225771 without member 1, the subsequence without 75 = 3*5^2, 147 = 3*7^2, 225 = (3*5)^2, 275 = 5^2*11, ... - Wolfdieter Lang, Jan 14 2025

Intersection of A005408 and A154777.

Cf. A033200 (primes), A033203, A225771.

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A296938 Rational primes that decompose in the field Q(sqrt(17)).

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

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A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

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A374294 a(n) is the smallest positive integer k such that A002325(k) = n.

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A201544 Odd numbers of the form x^2 + 2*y^2 with positive integers x and y.

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