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 1-4 of 4 results.

A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 6, 8, 8, 9, 10, 10, 8, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

References

  • A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002331, A002313, A002144.

Programs

  • Maple
    a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a),[p,x,y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"
  • Mathematica
    Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *)
  • PARI
    f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p, t=s;s=x%s;x=t);s
forprime(p=2,1e3,if(p%4-3,print1(f(p)", "))) \\ Charles R Greathouse IV, Apr 24 2012
  • PARI
    do(p)=qfbsolve(Qfb(1,0,1),p)[1]
forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013
  • PARI
    print1(1); forprimestep(p=5,1e3,4, print1(", "qfbcornacchia(1,p)[1])) \\ Charles R Greathouse IV, Sep 15 2021

Formula

a(n) = A096029(n) + A096030(n) + 1, for n>1. - Lekraj Beedassy, Jul 21 2004
a(n+1) = Max(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010

A005098 Numbers k such that 4k + 1 is prime.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 13, 15, 18, 22, 24, 25, 27, 28, 34, 37, 39, 43, 45, 48, 49, 57, 58, 60, 64, 67, 69, 70, 73, 78, 79, 84, 87, 88, 93, 97, 99, 100, 102, 105, 108, 112, 114, 115, 127, 130, 135, 139, 142, 144, 148, 150, 153, 154, 160, 163, 165, 168, 169, 175, 177, 183
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Sum of i-th and j-th triangular numbers, where i=A096029(n), j=A096030(n); i.e., a(n) = A000217(A096029(n)) + A000217(A096030(n)). - Lekraj Beedassy, Jun 16 2004
For every k in the sequence, there is exactly 1 square number that can be subtracted to leave a pronic (A002378). E.g., 27 - 25 = 2, 99 - 9 = 90. - Jon Perry, Nov 06 2010
See A208295 for details concerning the preceding Jon Perry comment. - Wolfdieter Lang, Mar 29 2012
a(k) appears in the o.g.f. for floor(A002144(k)*j^2/4), j >= 0, for k >= 1: x*(a(k)*(1 + x^2) + b(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k) + 1)/2 = A119681(k). - Wolfdieter Lang, Aug 07 2013

Links

Crossrefs

See A002144 for the actual primes.
Cf. A002972, A002973, A173331, A173330, A023212.

Programs

  • Haskell
    a005098 = (`div` 4) . (subtract 1) . a002144
-- Reinhard Zumkeller, Mar 17 2013
  • Magma
    [k: k in [0..10000] | IsPrime(4*k+1)] // Vincenzo Librandi, Nov 18 2010
  • Maple
    a := []; for k from 1 to 500 do if isprime(4*k+1) then a := [op(a), k]; fi; od: A005098 := k->a[k];
  • Mathematica
    Select[Range[200], PrimeQ[4# + 1] &] (* Harvey P. Dale, Apr 20 2011 *)
  • PARI
    is(k)=isprime(4*k+1) \\ Charles R Greathouse IV, Nov 20 2012

Formula

a(n) = (A002144(n)-1)/4.

Extensions

More terms from Ray Chandler, Jun 26 2004
Edited by Charles R Greathouse IV, Mar 17 2010

A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 5, 3, 5, 4, 1, 3, 7, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 4, 6, 15, 22, 11, 3, 5
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

References

  • A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002330, A002313, A002144, A027862 (locates y=x+1).

Programs

  • Maple
    See A002330 for Maple program.
# alternative
A002331 := proc(n)
    A363051(A002313(n)) ;
end proc:
seq(A002331(n),n=1..100) ; # R. J. Mathar, Feb 01 2024
  • Mathematica
    pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, pJean-François Alcover, Feb 26 2016 *)
  • PARI
    f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p,t=s;s=x%s;x=t);s
forprime(p=2,1e3,if(p%4-3,print1(sqrtint(p-f(p)^2)", ")))
\\ Charles R Greathouse IV, Apr 24 2012
  • PARI
    do(p)=qfbsolve(Qfb(1,0,1),p)[2]
forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013

Formula

a(n) = A096029(n) - A096030(n) for n > 1. - Lekraj Beedassy, Jul 16 2004
a(n+1) = Min(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A363051(A002313(n)). - R. J. Mathar, Jan 31 2024

A096029 Values (x+y-1)/2, where x^2+y^2=p runs over the Pythagorean primes A002144.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 6, 7, 7, 8, 8, 7, 9, 9, 7, 8, 10, 9, 8, 11, 11, 10, 9, 12, 12, 12, 11, 12, 12, 13, 12, 10, 11, 14, 14, 13, 12, 14, 13, 15, 15, 16, 16, 12, 15, 14, 17, 17, 14, 17, 15, 17, 13, 15, 18, 14, 17, 19, 18, 19, 18, 16, 19, 18, 20, 16, 17, 20, 21, 21, 19, 20
Offset: 1

Views

Author

Lekraj Beedassy, Jun 16 2004

Keywords

Crossrefs

Cf. A005098, A096030.

Formula

a(n)=(A079886(n) - 1)/2

Extensions

More terms from Ray Chandler, Jun 26 2004
Showing 1-4 of 4 results.

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

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