cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A139513 Primes congruent to {1, 3, 7, 9} mod 20.

Original entry on oeis.org

3, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 563, 569, 587, 601, 607, 641
Offset: 1

Views

Author

Artur Jasinski, Apr 25 2008

Keywords

Comments

Rational primes that decompose in the field Q(sqrt(-5)). - N. J. A. Sloane, Dec 25 2017

References

  • Dirichlet & Dedekind, Lectures on Number Theory (English Translation 1999), p. 119.
  • David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989, p. 14 (1.8), p. 32 (2.19).

Links

Crossrefs

Cf. A296922, A003626.

Programs

  • Magma
    [ p: p in PrimesUpTo(700) | p mod 20 in [1,3,7,9] ]; // Vincenzo Librandi, Aug 15 2012
  • Mathematica
    a = {}; Do[If[MemberQ[{1, 3, 7, 9}, Mod[Prime[n], 20]], AppendTo[a, Prime[n]]], {n, 1, 200}]; a (*Artur Jasinski*)
Select[Prime[Range[200]],MemberQ[{1,3,7,9},Mod[#,20]]&] (* Vincenzo Librandi, Aug 15 2012 *)
  • PARI
    select(p->my(k=p%20); k==1 || k==3 || k==7 || k==9, primes(100)) \\ Charles R Greathouse IV, Nov 29 2016

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016
Legendre symbol (-5, a(n)) = +1. One sets (-5, 5) = 0 and for odd primes p == -1, -3, -7, -9 (mod 20) (-5, p) = -1, given in A003626. - Wolfdieter Lang, Mar 05 2021

A091727 Norms of prime ideals of Z[sqrt(-5)].

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 121, 127, 149, 163, 167, 169, 181, 223, 227, 229, 241, 263, 269, 281, 283, 289, 307, 347, 349, 361, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487
Offset: 1

Views

Author

Paul Boddington, Feb 02 2004

Keywords

Comments

Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.
From Jianing Song, Feb 20 2021: (Start)
The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[sqrt(-5)] has class number 2.
For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.
For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.
For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)

Examples

			From _Jianing Song_, Feb 20 2021: (Start)
Let |I| be the norm of an ideal I, then:
|(2, 1+sqrt(-5))| = 2;
|(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3;
|(sqrt(-5))| = 5;
|(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7;
|(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23;
|(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29;
|(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End)

References

  • David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989.
  • A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991.

Links

Crossrefs

Cf. A091728.
Cf. A289741, A033205, A106865, A139513, A003626.
The number of distinct ideals with norm n is given by A035170.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), this sequence (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

Programs

  • PARI
    isA091727(n) = { my(ms = [1, 2, 3, 5, 7, 9], p, e=isprimepower(n,&p)); if(!e || e>2, 0, bitxor(e-1,!!vecsearch(ms,p%20))); }; \\ Antti Karttunen, Feb 24 2020

Extensions

Offset corrected by Jianing Song, Feb 20 2021

A122869 Primes p that divide Lucas((p-1)/2), where Lucas is A000032.

Original entry on oeis.org

11, 19, 31, 59, 71, 79, 131, 139, 151, 179, 191, 199, 211, 239, 251, 271, 311, 331, 359, 379, 419, 431, 439, 479, 491, 499, 571, 599, 619, 631, 659, 691, 719, 739, 751, 811, 839, 859, 911, 919, 971, 991, 1019, 1031, 1039, 1051, 1091, 1151, 1171, 1231, 1259
Offset: 1

Views

Author

Alexander Adamchuk, Sep 16 2006

Keywords

Comments

Final digit of a(n) is 1 or 9.
A002145 is the union of this sequence and A122870, Primes p that divide Lucas((p+1)/2).
Conjecture: This sequence is just the primes congruent to 11 or 19 mod 20. - Charles R Greathouse IV, May 25 2011 [The conjecture is correct. - Jianing Song, Jun 20 2025]
Note that F(p-1) = F((p-1)/2)*Lucas((p-1)/2), where F = A000045. Since gcd(F(n),Lucas(n)) = 1 or 2 (because Lucas(n)^2 - 5*F(n)^2 = 4*(-1)^n), this sequence lists primes p such that p divides F(p-1) but does not divides F((p-1)/2). By Propositions 1.1 and 1.2 (the k = 3 case) of my link below, this is primes p == 11, 19 (mod 20). - Jianing Song, Jun 20 2025

Links

Crossrefs

Cf. A000032, A000045, A053032, A076518, A122870.
Subsequence of A002145, A003626, A040105, A040147 and A064739.

Programs

  • Mathematica
    Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1-1)/2-1]+Fibonacci[(#1-1)/2+1])/#1]&]
  • PARI
    lista(kmax) = {my(lucas1 = 1, lucas2 = 3, lucas3, p); for(k = 3, kmax, lucas3 = lucas1 + lucas2; p = 2*k + 1; if(isprime(p) && !(lucas3 % p), print1(p, ", ")); lucas1 = lucas2; lucas2 = lucas3);} \\ Amiram Eldar, Jun 06 2024

A363416 a(n) = 1/sqrt(5) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-5).

Original entry on oeis.org

0, 1, 3, -24, -240, 1890, 40446, -311472, -12038544, 86898420, 5614173180, -36099955584, -3786960576672, 20307572439336, 3492389655843480, -14110473458954880, -4223754447793582464, 10493742733654512528, 6488421280167604253616, -4618066393756887442560, -12344309538368967592151040
Offset: 0

Views

Author

Peter Bala, Jun 01 2023

Keywords

Comments

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, 3, 7 or 9 (mod 20), equivalently, rational primes that split in the field extension Q(sqrt(-5)) of Q, together with the prime p = 2. See A139513.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/4 as n -> oo.
(ii) for the other primes 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 == 11, 13, 17 or 19 (mod 20), equivalently, primes that remain inert in the field extension Q(sqrt(-5)) of Q, together with the prime p = 5, which ramifies in Q(sqrt(-5)). See A003626.

Examples

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 1..80] = [0, 1, 1, 1, 3, 3, 3, 4, 4, 4, 5, 5, 5, 7, 7, 7, 8, 8, 8, 9, 9, 9, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 16, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22, 22, 25, 25, 25, 26, 26, 26, 27, 27, 27, 29, 29, 29, 30, 30, 30, 31, 31, 31, 33, 33, 33, 34, 34, 34, 35, 35, 35, 39, 39, 41, 40, 40].
Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 11: the sequence of 11-adic valuations [v_11(a(n)) : n = 1..121] = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Links

Crossrefs

Cf. A105750, A105751, A363409 - A363415

Programs

  • Maple
    a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Formula

P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n-2*k).

A296923 Primes p such that Legendre(-5,p) = -1.

Original entry on oeis.org

2, 11, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 211, 233, 239, 251, 257, 271, 277, 293, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 419, 431, 433, 439, 457, 479, 491, 499, 557, 571, 577, 593, 599, 613
Offset: 1

Views

Author

N. J. A. Sloane, Dec 25 2017

Keywords

Comments

Primes == 2, 11, 13, 17, or 19 (mod 20). - Robert Israel, Dec 27 2017

Links

Crossrefs

Cf. A296922, A003626.

Programs

  • Maple
    Load the Maple program HH given in A296920. Then run HH(-5,200);
select(isprime, {seq(seq(20*i+j,j=[2,11,13,17,19]),i=0..100)}); # Robert Israel, Dec 27 2017
  • PARI
    lista(nn) = forprime(p=2, nn, if (kronecker(-5,p) == -1, print1(p, ", "))); \\ Michel Marcus, Dec 26 2017

A363415 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-5).

Original entry on oeis.org

1, 1, -9, -54, 426, 6426, -50274, -1465884, 10992996, 552727476, -3792193524, -312571718424, 1853425616616, 248005863100296, -1173524207653224, -263102748395914224, 865735128320476176, 359884863190774985616, -584551982838131141904, -616984573598760535235424, -155177934223071790979424
Offset: 0

Views

Author

Peter Bala, Jun 01 2023

Keywords

Comments

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes. Numerical calculation suggests that a similar division also holds in this case.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {5, 11, 13, 31, 53, 79, 97, 113, 137, 157, 179, 193, 197, ...}.
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 begins {2, 3, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, ...} and consists of primes p == 1, 3, 7 or 9 (mod 20), equivalently, rational primes that split in the field extension Q(sqrt(-5)) of Q, together with the prime p = 2. See A139513.
It can be shown that the 2-adic valuation v_2(a(n)) = floor((n+1)/4).
Moll's conjecture 5.5 extends to this sequence: for type two primes p > 2, 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 begins {17, 19, 37, 59, 71, 73, 131, 151, 173, 191, 199, ...}.
Taken together, the type 1 and type 3 primes appear to consist of primes p == 11, 13, 17 or 19 (mod 20), equivalently, primes that remain inert in the field extension Q(sqrt(-5)) of Q, together with the prime p = 5, which ramifies in Q(sqrt(-5)). See A003626.

Examples

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 0..80] = [0, 0, 2, 3, 1, 3, 3, 3, 4, 4, 4, 5, 5, 5, 7, 7, 7, 8, 8, 8, 9, 9, 9, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 16, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22, 22, 25, 25, 25, 26, 26, 26, 27, 27, 27, 29, 29, 29, 30, 30, 30, 31, 31, 31, 33, 33, 33, 34, 34, 34, 35, 35, 35, 39, 39, 40, 40].
Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 17: the sequence of 17-adic valuations [v_17(a(n)) : n = 0..100] = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Links

Crossrefs

Cf. A105750, A363409 - A363416.

Programs

  • Maple
    a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Formula

a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (5*n + 5)*tan(Sum_{k = 1..n} arctan(sqrt(5)*k))/sqrt(5).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
a(0) = a(1) = 1.

A369863 Inert rational primes in the field Q(sqrt(-21)).

Original entry on oeis.org

13, 29, 43, 47, 53, 59, 61, 67, 73, 79, 83, 97, 113, 127, 131, 137, 149, 151, 157, 163, 167, 181, 197, 211, 227, 229, 233, 241, 251, 281, 311, 313, 317, 331, 349, 379, 383, 389, 397, 401, 409, 419, 433, 449, 463, 467, 479, 487, 499, 503, 547, 557, 563, 569, 571, 577, 587
Offset: 1

Views

Author

Dimitris Cardaris, Feb 03 2024

Keywords

Comments

Primes p such that Legendre(-21,p) = -1.

Crossrefs

Cf. inert rational primes in the imaginary quadratic field Q(sqrt(-d)) for the first squarefree positive integers d: A002145 (1), A003628 (2), A003627 (3), A003626 (5), A191059 (6), A003625 (7), A296925 (10), A191060 (11), A105885 (13), A191061 (14), A191062 (15), A296930 (17), A191063 (19), this sequence (21), A191064 (22), A191065 (23).

Programs

  • Mathematica
    Select[Range[3,600], PrimeQ[#] && JacobiSymbol[-21,#]==-1 &] (* Stefano Spezia, Feb 04 2024 *)
  • SageMath
    [p for p in prime_range(3, 600) if legendre_symbol(-21, p) == -1]
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