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 1-3 of 3 results.

A180471 Irregular triangle in which row n has all primes q such that prime(n)*q is a base-2 Fermat pseudoprime.

Original entry on oeis.org

31, 257, 73, 89, 683, 113, 11, 151, 331, 73, 109, 61681, 127, 337, 5419, 178481, 2796203, 157, 1613, 233, 1103, 2089, 3033169, 1321, 20857, 599479, 281, 86171, 122921, 19, 37, 109, 433, 38737, 2731, 8191, 121369, 22366891, 13367, 164511353, 8831418697, 23, 353, 397, 683, 2113, 2931542417
Offset: 5

Views

Author

T. D. Noe, Jan 19 2011

Keywords

Comments

The length of row n is A085014(n). The smallest and largest primes in row n are A085012(n) and A085019(n).

Examples

			The irregular triangle begins
31
none
257
73
89, 683
113
11, 151, 331
73, 109
61681

References

Links

Crossrefs

Cf. A085012, A085014, A085019.

Programs

  • Mathematica
    Flatten[Table[p=Prime[n]; q=Transpose[FactorInteger[2^(p-1)-1]][[1]]; cnt={}; Do[If[PowerMod[2, p*q[[i]]-1, p*q[[i]]]==1, AppendTo[cnt,q[[i]]]], {i,Length[q]}]; cnt, {n,5,50}]]

A083752 Minimal k > n such that (4k+3n)(4n+3k) is a square.

Original entry on oeis.org

393, 786, 1179, 109, 1965, 2358, 2751, 218, 3537, 3930, 4323, 327, 132, 5502, 5895, 436, 6681, 7074, 7467, 545, 8253, 8646, 9039, 157, 9825, 264, 10611, 763, 11397, 11790, 12183, 872, 481, 13362, 13755, 981, 184, 14934, 396, 1090, 16113, 16506, 16899, 1199
Offset: 1

Views

Author

Zak Seidov, Jun 17 2003

Keywords

Comments

A problem of elementary geometry lead to the search for squares of the form (4*a^2+3*b^2)(4*b^2+3*a^2). I could not find any such squares except when a=b. See link to ZS.
Letting j := 24k+25n in (4k+3n)(4n+3k)=x^2 yields the Pell-like equation j^2 - 48 x^2 = 49 n^2. The recurrence relationship for solutions to Pell equations implies that if k,x is a solution for n, then so is k1=18817k+19600n-5432x, x1=18817x-65184k-67900n. As a result, if there is a solution with 109/4n < k < 393n, then there is also one with n < k < 109/4n, so either n < a(n) <= 109/4n or a(n)=393n. - David Applegate, Jan 09 2014

Examples

			a(24)=157 because (4*157+3*24)(3*157+4*24)= 396900=630*630.

Links

Crossrefs

Cf. A085018, A085019.
Cf. A010052, A235188.

Programs

  • Haskell
    a083752 n = head [k | k <- [n+1..], a010052 (12*(k+n)^2 + k*n) == 1]
-- Reinhard Zumkeller, Apr 06 2015
  • Maple
    a:= proc(n) local k; for k from n+1
      while not issqr((4*k+3*n)*(4*n+3*k)) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 13 2013
  • Mathematica
    a[n_] := For[k = n + 1, True, k++, If[IntegerQ[Sqrt[(4k+3n)(4n+3k)]], Return[k]]]; Table[an = a[n]; Print[an]; an, {n, 1, 50}] (* Jean-François Alcover, Oct 31 2016 *)
  • PARI
    a(n)=my(k=n+1); while(!issquare((4*k+3*n)*(4*n+3*k)), k++); k \\ Charles R Greathouse IV, Dec 13 2013
  • PARI
    diff(v)=vector(#v-1,i,v[i+1]-v[i])
a(n)=my(v=select(k->issquare(12*Mod(k,n)^2),[0..n-1])); forstep(k=n+v[1], 393*n, diff(concat(v,n)), if(issquare((4*k+3*n)*(4*n+3*k)) && k>n, return(k))) \\ Charles R Greathouse IV, Dec 13 2013
  • PARI
    a(n)=for(k=n+1, 109*n\4, if(issquare((4*k+3*n)*(4*n+3*k)), return(k))); 393*n \\ Charles R Greathouse IV, Jan 09 2014
  • Sage
    def a(n):
    k = n + 1
    while not is_square((4*k+3*n)*(4*n+3*k)):
        k += 1
    return k
[a(n) for n in (1..44)] # Peter Luschny, Jun 25 2014

Formula

(4a(n)+3n)(4n+3a(n)) is a square.
n < a(n) <= 393n. - Charles R Greathouse IV, Dec 13 2013

Extensions

a(12) corrected by Charles R Greathouse IV, Dec 13 2013

A085018 Numbers n such that there is no divisor m of n with mA083752(n) = (n/m)A083752(m).

Original entry on oeis.org

1, 4, 13, 24, 33, 37, 52, 61, 69, 73, 88, 97, 109, 121, 132, 141, 157, 177, 181, 184, 193, 213, 229, 241, 244, 249, 253, 277, 292, 312, 313, 321, 337, 349, 373, 376, 388, 393, 397, 409, 421, 429, 433, 457, 472, 481, 501, 517, 529, 537, 541, 564, 568, 573, 577
Offset: 1

Views

Author

Zak Seidov, Jun 18 2003

Keywords

Comments

Seems to be a subsequence of the positive numbers primitively represented by the binary quadratic form (1, 6, -3) with discriminant 48 (see A244291, A243168). - Peter Luschny, Jun 25 2014

Examples

			A083752(2) = (2/1)*A083752(1), therefore 2 is not in the sequence.
But A083752(4) = 109 and 4*A083752(1) = 1572 and 2*A083752(2) = 1572.
Therefore the equation cannot be solved and 4 is in the sequence.

Links

Crossrefs

Cf. A083752, A085019, A244291, A243168.

Programs

  • Mathematica
    (* b = A083752 *) b[n_] := b[n] = For[k = n+1, True, k++, If[IntegerQ[Sqrt[(4k+3n)(4n+3k)]], Return[k]]]; Reap[For[n = 1, n < 600, n++, mm = Most @ Divisors[n]; If[NoneTrue[mm, b[n] == (n/#) b[#] &], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)
  • Sage
    def is_A085018(n):
    for d in divisors(n):
        if d < n:
            if d*A083752(n) == n*A083752(d):
                return false
    return true
filter(is_A085018, (1..577)) # Peter Luschny, Jun 25 2014

Extensions

Edited and extended by Stefan Steinerberger, Jul 30 2007
More terms from Peter Luschny, Jun 25 2014
Showing 1-3 of 3 results.

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