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.

Previous Showing 11-17 of 17 results.

A296937 Rational primes that decompose in the field Q(sqrt(13)).

Original entry on oeis.org

3, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Comments

Is this the same sequence as A141188 or A038883? - R. J. Mathar, Jan 02 2018
From Jianing Song, Apr 21 2022: (Start)
Primes p such that Kronecker(13, p) = Kronecker(p, 13) = 1, where Kronecker() is the Kronecker symbol. That is to say, primes p that are quadratic residues modulo 13.
Primes p such that p^6 == 1 (mod 13).
Primes p == 1, 3, 4, 9, 10, 12 (mod 13). (End)

Links

Crossrefs

Cf. A011583 (kronecker symbol modulo 13), A038883.
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), this sequence (D=13), A296938 (D=17).
Cf. A038884 (inert rational primes in the field Q(sqrt(13))).

Programs

Formula

Equals A038883 \ {13}. - Jianing Song, Apr 21 2022

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Comments

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)

Links

Crossrefs

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))).

Programs

A365313 Prime powers (A246655) q such that 3 is a nonzero square in the finite field F_q.

Original entry on oeis.org

11, 13, 23, 25, 37, 47, 49, 59, 61, 71, 73, 83, 97, 107, 109, 121, 131, 157, 167, 169, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 289, 311, 313, 337, 347, 349, 359, 361, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 529, 541, 563
Offset: 1

Views

Author

Jianing Song, Oct 22 2023

Keywords

Comments

Prime powers q that are congruent to 1 or 11 modulo 12 (see A366526).
Odd prime powers q such that 3^((q-1)/2) = 1 in F_q.
Prime powers q such that x^2 - 3 splits into different linear factors in F_q[x].
Contains the powers of primes congruent to 1 or 11 modulo 12 and the even powers of primes congruent to 5 or 7 modulo 12.

Examples

			49 is a term since 3 = -4 = (+-2i)^2 in F_49 = F_7(i).

Links

Crossrefs

Supersequence of A097933.
Prime powers q such that a is a nonzero square in F_q: A365082 (q=-2), A085759 (q=-1), A366526 (q=2), this sequence (q=3).

Programs

  • PARI
    isA365313(n) = isprimepower(n) && (n%12==1 || n%12==11)

A164624 Primes p such that p + Floor[p/2] + Floor[p/3] is prime.

Original entry on oeis.org

2, 3, 11, 13, 23, 37, 59, 83, 109, 131, 181, 191, 193, 229, 311, 337, 373, 383, 397, 443, 479, 541, 563, 599, 613, 743, 769, 877, 911, 983, 1019, 1021, 1031, 1091, 1093, 1129, 1153, 1163, 1237, 1283, 1297, 1319, 1381, 1451, 1453, 1489, 1523, 1559, 1571, 1597
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Keywords

Comments

11+5+3=19, 13+6+4=23, 23+11+7=41, ..

Crossrefs

Subsequence of A038874. Cf. A038874, A097933

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+Floor[p/2]+Floor[p/3]],AppendTo[lst,p]],{n,2*6!}];lst

Extensions

Edited by Charles R Greathouse IV, Nov 02 2009

A164625 Primes p such that p+floor(p/2)+floor(p/3)+floor(p/5) is also prime.

Original entry on oeis.org

2, 3, 7, 19, 83, 89, 127, 137, 139, 181, 251, 257, 311, 317, 373, 379, 449, 491, 499, 503, 509, 673, 733, 797, 853, 857, 863, 919, 971, 983, 1033, 1039, 1049, 1093, 1151, 1201, 1217, 1399, 1453, 1579, 1583, 1627, 1697, 1741, 1871, 1933, 1993, 2129, 2237, 2281
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Keywords

Examples

			For p=7, 7+3+2+1=13 is prime, which admits 7=a(4) to the sequence.
For p=19, 19+9+6+3=37 is prime, which puts 19=a(5) into the sequence.

Links

Crossrefs

Cf. A038874, A049545, A097933, A164624

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+Floor[p/2]+Floor[p/3]+Floor[p/5]],AppendTo[lst, p]],{n,2*6!}];lst
Select[Prime[Range[350]],PrimeQ[Total[Floor[#/{2,3,5}]]+#]&] (* Harvey P. Dale, Feb 19 2012 *)

Extensions

Comments rephrased as examples by R. J. Mathar, Aug 20 2009

A296933 Primes p such that Legendre(3,p) = 0 or 1.

Original entry on oeis.org

3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Links

Crossrefs

This is A038874 without the initial 2.
Cf. A003630, A038875, A097933, A296920.

Programs

  • Maple
    # Load the Maple program HH given in A296920. Then run HH(3, 200); This produces A097933, A003630, this sequence, and A038875.
  • Mathematica
    Join[{3}, Select[Prime[Range[200]], JacobiSymbol[3, #] == 1 &]] (* Paolo Xausa, May 11 2024 *)

A334614 a(n) = pi(prime(n) - n) + n, where pi is the prime counting function.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 11, 13, 15, 18, 19, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 45, 47, 48, 50, 51, 53, 55, 57, 60, 61, 65, 66, 67, 68, 70, 72, 74, 76, 77, 79, 81, 82, 85, 88, 89, 91, 93, 94, 95, 99, 101, 102, 104, 105, 106, 107, 108, 112, 116, 117
Offset: 1

Views

Author

Ya-Ping Lu, Sep 08 2020

Keywords

Comments

It can be shown that a(n) > a(n-1) >= 1 and a(n) <= 2*n - 1 < 2*n (see proofs in the Links section).

Links

Crossrefs

Cf. A000040, A000720, A014688, A014689, A062298, A065328, A097933.
Cf. A332086, A337334.

Programs

  • Mathematica
    Table[PrimePi[Prime[n] - n] + n, {n, 1, 64}] (* Amiram Eldar, Sep 09 2020 *)
  • PARI
    a(n) = n + primepi(prime(n) - n); \\ Michel Marcus, Sep 09 2020
  • Python
    from sympy import prime, primepi
for n in range(1, 100001):
    a_n = primepi(prime(n) - n) + n
    print(a_n)

Formula

a(n) = A000720(A014689(n)) + n.
a(n) = A065328(n) + n. - Michel Marcus, Sep 12 2020
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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