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 1-5 of 5 results.

A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 13, 15, 16, 18, 21, 22, 24, 25, 27, 28, 31, 34, 36, 37, 39, 42, 43, 45, 46, 48, 49, 51, 55, 57, 58, 60, 64, 66, 67, 69, 70, 72, 73, 76, 78, 79, 81, 84, 87, 88, 91, 93, 94, 97, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 120, 121, 123, 126, 127
Offset: 1

Views

Author

Lekraj Beedassy, May 06 2004

Keywords

Comments

For the actual numbers 4n+1, see A008846(n).
Complement of A124934; A125203(a(n)) = 0; A000290 and A000217 are subsequences. - Reinhard Zumkeller, Nov 24 2006

Links

Programs

  • Haskell
    import Data.List (elemIndices)
a094178 n = a094178_list !! (n-1)
a094178_list = map (+ 1) $ elemIndices 0 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

Extensions

More terms from Ray Chandler, Jun 20 2004

A124934 Numbers of the form 4mn - m - n, where m, n are positive integers.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 26, 29, 30, 32, 33, 35, 38, 40, 41, 44, 47, 50, 52, 53, 54, 56, 59, 61, 62, 63, 65, 68, 71, 74, 75, 77, 80, 82, 83, 85, 86, 89, 90, 92, 95, 96, 98, 101, 103, 104, 107, 109, 110, 113, 116, 117, 118, 119, 122, 124, 125, 128, 129, 131
Offset: 1

Views

Author

Nick Hobson, Nov 13 2006

Keywords

Comments

a(n) misses the squares since (2x)^2 + 1 = (4m - 1)(4n - 1) is impossible.
a(n) misses the triangular numbers since (2x + 1)^2 + 1 = 2(4m - 1)(4n - 1) is impossible.
Taking m = k(k - 1)/2, n = k(k + 1)/2 gives 4mn - m - n = (k^2 - 1)^2 - 1, so a(n) is one less than a square infinitely often.
Complement of A094178; A125203(a(n)) > 0; union of A125217 and A125218; range of A125199. - Reinhard Zumkeller, Nov 24 2006

Examples

			a(1) = 2 because 2 = 4*1*1 - 1 - 1 is the smallest value in the sequence.

References

  • L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. Dover Publications, Inc., Mineola, NY, 2005, p. 401.

Links

Crossrefs

Cf. A000290, A000217.

Programs

  • Haskell
    import Data.List (findIndices)
a124934 n = a124934_list !! (n-1)
a124934_list = map (+ 1) $ findIndices (> 0) a125203_list
-- Reinhard Zumkeller, Jan 02 2013

Extensions

More terms from Reinhard Zumkeller, Nov 24 2006

A193773 Number of ways to write n as 2*x*y - x - y with 1 <= x <= y.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 3, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 4, 1, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 1, 2, 4, 1, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 4, 1, 2, 3
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 02 2013

Keywords

Comments

a(A005097(n)) = 1; for n > 1: a(A047845(n)) > 1. - Reinhard Zumkeller, Jan 02 2013
Number of ways to write 2*n+1 as a difference of two squares. Note that 2*(2*x*y - x - y) + 1 = (2*x - 1) * (2*y - 1) = (y + x - 1)^2 - (y - x)^2. - Michael Somos, Dec 23 2018

Examples

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... - _Michael Somos_, Dec 23 2018

Links

Crossrefs

Cf. A000005, A005097, A047845, A125203.

Programs

  • Haskell
    a193773 n = length [() | x <- [1 .. n + 1],
                         let (y,m) = divMod (x + n) (2 * x - 1),
                         x <= y, m == 0]
  • Mathematica
    a[ n_] := If[ n < 0, 0, Ceiling[ DivisorSigma[0, 2 n + 1] / 2]]; (* Michael Somos, Dec 23 2018 *)
  • PARI
    {a(n) = if(n < 0, 0, (numdiv(2*n+1) + 1)\2)}; /* Michael Somos, Dec 23 2018 */

Formula

a(n) = ceiling(A000005(2*n+1) / 2). - Michael Somos, Dec 23 2018

A125217 Numbers that can be written uniquely as 4*x*y-x-y with 1<=x<=y.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 29, 30, 32, 33, 35, 38, 40, 44, 50, 52, 53, 54, 59, 61, 62, 63, 65, 75, 77, 80, 82, 83, 85, 90, 92, 95, 98, 103, 104, 109, 113, 117, 118, 119, 122, 124, 125, 129, 132, 134, 137, 138, 143, 145, 147, 149, 151, 158, 159, 162, 164, 167
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 24 2006

Keywords

Comments

A125203(a(n)) = 1.
A124934 is the union of this sequence and A125218.

Links

Programs

  • Haskell
    import Data.List (elemIndices)
a125217 n = a125217_list !! (n-1)
a125217_list = map (+ 1) $ elemIndices 1 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

A125218 Numbers having at least two representations as 4*x*y-x-y with 1<=x<=y.

Original entry on oeis.org

26, 41, 47, 56, 68, 71, 74, 86, 89, 96, 101, 107, 110, 116, 128, 131, 140, 146, 152, 155, 161, 166, 173, 176, 182, 185, 191, 194, 201, 206, 209, 215, 221, 224, 236, 239, 242, 250
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 24 2006

Keywords

Comments

A125203(a(n)) > 1.
A124934 is the union of this sequence and A125217.

Links

Programs

  • Haskell
    import Data.List (findIndices)
a125218 n = a125218_list !! (n-1)
a125218_list = map (+ 1) $ findIndices (> 1) a125203_list
-- Reinhard Zumkeller, Jan 02 2013
Showing 1-5 of 5 results.

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