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 1-7 of 7 results.

A259602 (A259260(n) + A259260(n+1)) / 2.

Original entry on oeis.org

4, 9, 16, 25, 16, 4, 9, 16, 25, 36, 25, 16, 25, 36, 49, 64, 81, 100, 64, 16, 25, 36, 49, 64, 81, 64, 36, 49, 64, 81, 100, 81, 64, 81, 100, 121, 144, 169, 196, 144, 64, 100, 144, 81, 36, 49, 64, 81, 64, 49, 64, 81, 100, 121, 144, 169, 144, 121, 100, 64, 81
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 30 2015

Keywords

Comments

All terms are squares by definition of A259260.

Links

Crossrefs

Cf. A259260, A000290, A086517, A259603, A259604, A259605.

Programs

  • Haskell
    a259602 n = a259602_list !! (n-1)
a259602_list = zipWith ((flip div 2 .) . (+))
                       a259260_list $ tail a259260_list

A259526 Smallest m such that A259260(m) = 2*n-1.

Original entry on oeis.org

1, 6, 7, 2, 12, 3, 8, 20, 21, 9, 4, 13, 45, 14, 5, 10, 22, 27, 28, 23, 11, 50, 15, 46, 42, 47, 16, 51, 33, 24, 29, 61, 60, 30, 25, 34, 52, 17, 48, 41, 88, 105, 49, 18, 53, 35, 26, 31, 67, 62, 63, 68, 32, 58, 36, 54, 19, 91, 106, 87, 111, 94, 107, 90, 120, 55
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 29 2015

Keywords

Comments

Sequence is defined for all numbers iff A259260 is a permutation of the odd numbers;
A259260(a(n)) = 2*n-1.

Links

Crossrefs

Cf. A259260.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a259526 = (+ 1) . fromJust . (`elemIndex` a259260_list) . subtract 1 . (* 2)

A086517 a(1) = 1 and then the smallest odd number not included earlier such that the arithmetic mean of a pair of successive terms is prime.

Original entry on oeis.org

1, 3, 7, 15, 11, 23, 35, 27, 19, 39, 43, 31, 51, 55, 63, 59, 47, 71, 75, 67, 79, 87, 91, 103, 99, 95, 83, 111, 107, 119, 135, 127, 147, 115, 139, 123, 131, 143, 155, 159, 167, 179, 183, 151, 163, 171, 175, 187, 195, 191, 203, 219, 227, 231, 215, 207, 239, 243, 211
Offset: 1

Views

Author

Amarnath Murthy, Jul 30 2003

Keywords

Comments

Second term onwards rearrangement of odd numbers of the type 4n+3.

Links

Crossrefs

Cf. A086518.
Cf. A259565, A259260, A259429, A259542, A010051, A005408.

Programs

  • Haskell
    import Data.List (delete)
a086517 n = a086517_list !! (n-1)
a086517_list = 1 : f 1 [3, 5 ..] where
   f x zs = g zs where
     g (y:ys) = if a010051' ((x + y) `div` 2) == 1
                   then y : f y (delete y zs) else g ys
-- Reinhard Zumkeller, Jun 30 2015
  • PARI
    v=[1];n=1;while(n<100,s=(n+v[#v])/2;if(type(s)=="t_INT",if(isprime(s)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0));n++);v \\ Derek Orr, Jun 16 2015

Extensions

More terms from David Wasserman, Mar 10 2005

A259429 With a(1) = 1, a(n) is the smallest number not already in the sequence such that the arithmetic mean of two neighboring terms is a cube.

Original entry on oeis.org

1, 15, 39, 89, 161, 271, 415, 17, 37, 91, 159, 273, 413, 19, 35, 93, 157, 275, 411, 21, 33, 95, 155, 277, 409, 23, 31, 97, 153, 279, 407, 25, 29, 99, 151, 281, 405, 27, 101, 149, 283, 403, 621, 65, 63, 187, 245, 5, 11, 43, 85, 165, 267, 419, 13, 3, 51, 77, 173, 259, 427, 597, 861, 163, 87, 41, 209, 223, 463, 561, 125
Offset: 1

Views

Author

Derek Orr, Jun 26 2015

Keywords

Comments

Believed to be a permutation of the odd integers.
A259603(n) = (a(n) + a(n+1)) / 2; a(A259537(n)) = 2*n-1. - Reinhard Zumkeller, Jun 30 2015
The scatter-plot shows interesting helix-like lenticular structures. - Antti Karttunen, May 29 2016

Links

Crossrefs

Cf. A259260, A086517, A175428.
Cf. A010057, A005408, A259537.
Cf. A259603, A259565, A259542.

Programs

  • Haskell
    import Data.List (delete)
a259429 n = a259429_list !! (n-1)
a259429_list = 1 : f 1 [3, 5 ..] where
   f x zs = g zs where
     g (y:ys) = if a010057 ((x + y) `div` 2) == 1
                   then y : f y (delete y zs) else g ys
-- Reinhard Zumkeller, Jun 29 2015
  • Mathematica
    a = {1}; Do[k = 1; While[Or[MemberQ[a, k], !IntegerQ@ Power[Mean[{a[[i - 1]], k}], 1/3]], k++]; AppendTo[a, k], {i, 2, 120}]; a (* Michael De Vlieger, May 29 2016 *)
  • PARI
    v=[1]; n=1; while(#v<200, s=(n+v[#v])/2; if(type(s)=="t_INT", if(ispower(s,3)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0)); n++); v

A259542 a(1) = 1, for n > 1 a(n) = smallest number not already in the sequence such that the arithmetic mean of two neighboring terms is a triangular number.

Original entry on oeis.org

1, 5, 7, 13, 17, 3, 9, 11, 19, 23, 33, 39, 51, 21, 35, 37, 53, 57, 15, 27, 29, 43, 47, 25, 31, 41, 49, 61, 71, 85, 97, 59, 73, 83, 99, 111, 45, 65, 67, 89, 93, 63, 69, 87, 95, 115, 125, 147, 159, 81, 75, 107, 103, 79, 77, 55, 101, 109, 131, 141, 165, 177
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 30 2015

Keywords

Comments

A259604(n) = (a(n) + a(n+1)) / 2;
conjecture: sequence is a permutation of the odd numbers, see also A259260, A259429;
a(A259543(n)) = 2*n-1.

Links

Crossrefs

Cf. A259260, A259429, A010054, A005408, A259604.

Programs

  • Haskell
    import Data.List (delete)
a259542 n = a259542_list !! (n-1)
a259542_list = 1 : f 1 [3, 5 ..] where
   f x zs = g zs where
     g (y:ys) = if a010054 ((x + y) `div` 2) == 1
                   then y : f y (delete y zs) else g ys

A259565 a(1) = 1, for n > 1 a(n) = smallest number not already in the sequence such that the arithmetic mean of two neighboring terms is a squarefree number.

Original entry on oeis.org

1, 3, 7, 5, 9, 11, 15, 13, 17, 21, 23, 19, 25, 27, 31, 29, 33, 35, 39, 37, 41, 43, 49, 45, 47, 55, 51, 59, 57, 53, 61, 63, 67, 65, 69, 71, 75, 73, 81, 77, 79, 85, 87, 83, 89, 93, 95, 91, 97, 105, 99, 103, 101, 109, 111, 107, 113, 115, 121, 117, 119, 125, 129
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 30 2015

Keywords

Comments

A259605(n) = (a(n) + a(n+1)) / 2;
conjecture: sequence is a permutation of the odd numbers;
a(A259570(n)) = 2*n-1.

Links

Crossrefs

Cf. A086517, A259260, A259429, A259542, A008966, A005117, A005408, A259570.
Cf. A259605.

Programs

  • Haskell
    import Data.List (delete)
a259565 n = a259565_list !! (n-1)
a259565_list = 1 : f 1 [3, 5 ..] where
   f x zs = g zs where
     g (y:ys) = if a008966 ((x + y) `div` 2) == 1
                   then y : f y (delete y zs) else g ys

A259603 a(n) = (A259429(n) + A259429(n+1)) / 2.

Original entry on oeis.org

8, 27, 64, 125, 216, 343, 216, 27, 64, 125, 216, 343, 216, 27, 64, 125, 216, 343, 216, 27, 64, 125, 216, 343, 216, 27, 64, 125, 216, 343, 216, 27, 64, 125, 216, 343, 216, 64, 125, 216, 343, 512, 343, 64, 125, 216, 125, 8, 27, 64, 125, 216, 343, 216, 8, 27
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 30 2015

Keywords

Comments

All terms are cubes by definition of A259260.

Links

Crossrefs

Cf. A259429, A000578, A086517, A259260, A259602, A259604, A259605.

Programs

  • Haskell
    a259603 n = a259603_list !! (n-1)
a259603_list = zipWith ((flip div 2 .) . (+))
                       a259429_list $ tail a259429_list
Showing 1-7 of 7 results.

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