A259604 -id:A259604 - cp's OEIS frontend

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.

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

Reinhard Zumkeller, Jun 30 2015

nonn

 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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

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

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

Reinhard Zumkeller, Jun 30 2015

nonn

All terms are squares by definition of A259260.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

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

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

Reinhard Zumkeller, Jun 30 2015

nonn

All terms are cubes by definition of A259260.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

a259603 n = a259603_list !! (n-1)
a259603_list = zipWith ((flip div 2 .) . (+))
                       a259429_list $ tail a259429_list

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

Reinhard Zumkeller, Jun 30 2015

nonn

All terms are squarefree by definition of A259565.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

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

A086518 Primes arising as the arithmetic mean of a pair of successive terms of A086517.

Original entry on oeis.org

2, 5, 11, 13, 17, 29, 31, 23, 29, 41, 37, 41, 53, 59, 61, 53, 59, 73, 71, 73, 83, 89, 97, 101, 97, 89, 97, 109, 113, 127, 131, 137, 131, 127, 131, 127, 137, 149, 157, 163, 173, 181, 167, 157, 167, 173, 181, 191, 193, 197, 211, 223, 229, 223, 211, 223, 241, 227, 223

Offset: 1

Amarnath Murthy, Jul 30 2003

nonn

			2 = (1+3)/2, 11 = (7+15)/2, etc.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A086517, A259602, A259603, A259604, A259605.

a086518 n = a086518_list !! (n-1)
a086518_list = zipWith ((flip div 2 .) . (+))
                       a086517_list $ tail a086517_list
-- Reinhard Zumkeller, Jun 30 2015

v=[1]; n=1; while(n<100, s=(n+v[#v])/2; if(type(s)=="t_INT", if(isprime(s)&&!vecsearch(vecsort(v), n), print1(s,", "); v=concat(v, n); n=0)); n++) \\ Derek Orr, Jun 16 2015

a(n) = (A086517(n)+A086517(n+1))/2. - David Wasserman, Mar 10 2005

More terms from David Wasserman, Mar 10 2005

cp's OEIS Frontend

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.

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

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

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

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Haskell

A086518 Primes arising as the arithmetic mean of a pair of successive terms of A086517.

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