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.

A259605 (A259565(n) + A259565(n+1)) / 2.

Original entry on oeis.org

2, 5, 6, 7, 10, 13, 14, 15, 19, 22, 21, 22, 26, 29, 30, 31, 34, 37, 38, 39, 42, 46, 47, 46, 51, 53, 55, 58, 55, 57, 62, 65, 66, 67, 70, 73, 74, 77, 79, 78, 82, 86, 85, 86, 91, 94, 93, 94, 101, 102, 101, 102, 105, 110, 109, 110, 114, 118, 119, 118, 122, 127
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 30 2015

Keywords

Comments

All terms are squarefree by definition of A259565.

Links

Crossrefs

Cf. A259565, A005117, A086517, A259602, A259603, A259604.

Programs

  • Haskell
    a259605 n = a259605_list !! (n-1)
a259605_list = zipWith ((flip div 2 .) . (+))
                       a259565_list $ tail a259565_list

A259570 Smallest m such that A259565(m) = 2*n-1.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 8, 7, 9, 12, 10, 11, 13, 14, 16, 15, 17, 18, 20, 19, 21, 22, 24, 25, 23, 27, 30, 26, 29, 28, 31, 32, 34, 33, 35, 36, 38, 37, 40, 41, 39, 44, 42, 43, 45, 48, 46, 47, 49, 51, 53, 52, 50, 56, 54, 55, 57, 58, 60, 61, 59, 65, 62, 67, 63, 64, 68
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 30 2015

Keywords

Comments

Sequence is a permutation of the natural numbers iff A259565 is a permutation of the odd numbers;
A259565(a(n)) = 2*n-1.

Links

Crossrefs

Cf. A259565.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a259570 = (+ 1) . fromJust . (`elemIndex` a259565_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

A259260 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 square.

Original entry on oeis.org

1, 7, 11, 21, 29, 3, 5, 13, 19, 31, 41, 9, 23, 27, 45, 53, 75, 87, 113, 15, 17, 33, 39, 59, 69, 93, 35, 37, 61, 67, 95, 105, 57, 71, 91, 109, 133, 155, 183, 209, 79, 49, 151, 137, 25, 47, 51, 77, 85, 43, 55, 73, 89, 111, 131, 157, 181, 107, 135, 65, 63, 99, 101, 141, 147, 191, 97, 103, 139, 149, 189, 203, 247, 145
Offset: 1

Views

Author

Derek Orr, Jun 22 2015

Keywords

Comments

Conjectured to be a permutation of the odd numbers.
A259602(n) = (a(n) + a(n+1)) / 2; a(A259526(n)) = 2*n-1. - Reinhard Zumkeller, Jun 29 2015

Links

Crossrefs

Cf. A034175, A086517.
Cf. A010052, A005408.
Cf. A259602, A259565, A259429, A259542.

Programs

  • Haskell
    import Data.List (delete)
a259260 n = a259260_list !! (n-1)
a259260_list = 1 : f 1 [3, 5 ..] where
   f x zs = g zs where
     g (y:ys) = if a010052 ((x + y) `div` 2) == 1
                   then y : f y (delete y zs) else g ys
-- Reinhard Zumkeller, Jun 29 2015
  • Mathematica
    s={1}; Do[n = Last@ s; a=2; While[(b = 2*a^2 - n) <= 0 || MemberQ[s, b], a++]; AppendTo[s, b], {100}]; s (* Giovanni Resta, Jun 23 2015 *)
  • PARI
    v=[1];n=1;while(#v<100,s=(n+v[#v])/2;if(type(s)=="t_INT",if(issquare(s)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0));n++);v

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
Showing 1-5 of 5 results.

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

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