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.

User: Salvador Cerdá

Salvador Cerdá's wiki page.

Salvador Cerdá has authored 3 sequences.

A277913 Nonsquare numbers n for which the smallest y>0 solution of the Pellian equation x^2 - n*y^2 = 1 divides n.

Original entry on oeis.org

2, 3, 6, 8, 12, 15, 20, 24, 30, 35, 42, 48, 56, 60, 63, 68, 72, 75, 78, 80, 84, 87, 90, 99, 110, 120, 132, 143, 156, 168, 180, 182, 195, 210, 224, 240, 248, 255, 264, 272, 288, 306, 312, 318, 323, 330, 336, 342, 360, 380, 399, 420, 440, 462, 483, 506, 528, 552, 564, 575, 588
Offset: 1

Views

Author

Salvador Cerdá, Nov 16 2016

Keywords

Examples

			2 is in the sequence because A002349(2)=2 divides 2.
180 is in the sequence because A002349(180)=12 divides 180.

Crossrefs

See A002349 for the smallest y>0 solution of x^2 - n*y^2 = 1.

Programs

  • Mathematica
    PellSolve[(m_Integer)?Positive] :=
  Module[{cf, n, s}, cof = ContinuedFraction[Sqrt[m]];
   n = Length[Last[cof]]; If[OddQ[n], n = 2*n];
   s = FromContinuedFraction[
     ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
f[n_] := If[! IntegerQ[Sqrt[n]], PellSolve[n][[2]]];
Select[Range[250], Mod[#, f[#]] == 0 &]

Formula

Lim_{n->infinity} a(n+1)/a(n) = 1 (conjectured).

Extensions

More terms from Michel Marcus, Dec 04 2016

A273869 Integers n such that floor(sqrt(n!)) (A055226(n)) is a prime number.

Original entry on oeis.org

3, 11, 14, 53, 110, 216, 322, 364, 389
Offset: 1

Views

Author

Salvador Cerdá, Jun 01 2016

Keywords

Comments

a(10) (if it exists) requires n > 4800.
No further terms <= 15000. - Eric M. Schmidt, Jun 05 2017

Examples

			3 is in the sequence because floor(sqrt(3!)) = 2 is prime.
11 is in the sequence because floor(sqrt(11!)) = 6317 is prime.
14 is in the sequence because floor(sqrt(14!)) = 295259 is prime.
4 is not in the sequence because floor(sqrt(4!)) = 2^2.

Crossrefs

Cf. A055226.

Programs

  • Mathematica
    Select[Table[n, {n, 1, 2500}], PrimeQ[Floor[Sqrt[#!]]] &]
  • PARI
    isok(n) = isprime(sqrtint(n!)); \\ Michel Marcus, Jun 10 2016
  • PARI
    lista(nn) = for(n=1, nn, if(ispseudoprime(sqrtint(n!)), print1(n, ", "))); \\ Altug Alkan, Jul 09 2016

A273932 Integers m such that ceiling(sqrt(m!)) is prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 21, 2132, 3084, 9301
Offset: 1

Views

Author

Salvador Cerdá, Jun 04 2016

Keywords

Comments

This sequence includes the known solutions of Brocard's problem as of 2016 (see A146968).

Examples

			3 is a term because 3! = 6, sqrt(6) = 2.449489742783178..., the ceiling of which is 3, which is prime.
4 is a term because 4! = 24, sqrt(24) = 4.898979485566356..., the ceiling of which is 5, which is prime.

Crossrefs

Cf. A055228 (ceiling(sqrt(n!))), A146968.

Programs

  • Mathematica
    Select[Range[3200], PrimeQ[Ceiling[Sqrt[#!]]] &]
  • Python
    from math import isqrt, factorial
from itertools import count, islice
from sympy import isprime
def A273932_gen(): # generator of terms
    return filter(lambda n:isprime(1+isqrt(factorial(n)-1)),count(1))
A273932_list = list(islice(A273932_gen(),7)) # Chai Wah Wu, Jul 29 2022

Extensions

a(9) from Giovanni Resta, Jun 20 2016

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