A072062 -id:A072062 - cp's OEIS frontend

A072061 [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5))/2 and []=floor.

1, 2, 3, 5, 4, 7, 6, 10, 8, 13, 9, 15, 11, 18, 12, 20, 14, 23, 16, 26, 17, 28, 19, 31, 21, 34, 22, 36, 24, 39, 25, 41, 27, 44, 29, 47, 30, 49, 32, 52, 33, 54, 35, 57, 37, 60, 38, 62, 40, 65, 42, 68, 43, 70, 45, 73, 46, 75, 48, 78, 50, 81, 51, 83, 53, 86, 55, 89, 56, 91, 58, 94

Offset: 1

Clark Kimberling, Jun 11 2002, Aug 17 2007

The same sequence can be defined as follows: "a(1) = 1 and, for n>1, a(n) = a(n-1) + n/2 if n is even, otherwise a(n) = smallest positive integer which has not yet appeared in the sequence." This was originally a separate entry in the database, contributed by John W. Layman, Jul 08 2004. Antti Karttunen noticed on Jul 10 2004 that the two entries appeared to be identical. This was finally proved by Clark Kimberling, Aug 22 2007.
A permutation of the positive integers. Bisections are the lower and upper Wythoff sequences.
The consecutive pairs (1,2), (3,5), (4,7), (6,10), ... are the much-studied Wythoff pairs, arising in connection with Wythoff's game.
Conjecture: For even n, the ratio a(n)/a(n-1) is asymptotic to (1 + sqrt(5))/2 as n becomes large. (At n=3000, the ratio is 1.61804697, compared to the exact value 1.61803399.) - John W. Layman, Jul 08 2004
A more general conjecture may be stated as follows: Define {a(n)} by a(1)=1 and, for n>1, a(n) = a(n-1)+floor(kn) if n is even, else a(n)=smallest positive integer which has not yet appeared in the sequence, where k is a positive real number. Then a(2n)/a(2n-1) is asymptotic to k+sqrt(k^2+1) for large n. - John W. Layman, Jul 08 2004

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

Eric Weisstein's World of Mathematics, Wythoff's game.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000201, A001950, A026272, A072062, A094077.

[n*(1+(-1)^n)/4+Floor((2*n+1-(-1)^n)*(1+Sqrt(5))/8) : n in [1..100]]; // Wesley Ivan Hurt, Apr 10 2015

A072061:=n->n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8): seq(A072061(n), n=1..100); # Wesley Ivan Hurt, Apr 10 2015

Table[n*(1 + (-1)^n)/4 + Floor[(2 n + 1 - (-1)^n) (1 + Sqrt[5])/8], {n, 100}] (* Wesley Ivan Hurt, Apr 10 2015 *)

lista(nn) = {v = []; for (n=1, nn, v = concat(v, nt = floor(n*(1+sqrt(5))/2)); v = concat(v, n+nt);); v;} \\ Michel Marcus, Apr 14 2015

a(n) = n*(1+(-1)^n)/4+floor((2*n+1-(-1)^n)*(1+sqrt(5))/8). - Wesley Ivan Hurt, Apr 10 2015

Edited by N. J. A. Sloane, Jul 26 2008

A243352 If n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2k-1; otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2k.

Original entry on oeis.org

1, 3, 5, 2, 7, 9, 11, 4, 6, 13, 15, 8, 17, 19, 21, 10, 23, 12, 25, 14, 27, 29, 31, 16, 18, 33, 20, 22, 35, 37, 39, 24, 41, 43, 45, 26, 47, 49, 51, 28, 53, 55, 57, 30, 32, 59, 61, 34, 36, 38, 63, 40, 65, 42, 67, 44, 69, 71, 73, 46, 75, 77, 48, 50, 79, 81, 83, 52, 85, 87, 89

Offset: 1

Antti Karttunen, Jun 04 2014

Odd numbers occur (in order) at the positions given by squarefree numbers, A005117, and even numbers occur (in order) at the positions given by their complement, nonsquarefree numbers, A013929.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A088610. Cf. A243343, A072062.

(define (A243352 n) (if (zero? (A008966 n)) (* 2 (A057627 n)) (+ (* 2 (A013928 n)) 1)))

If mu(n) = 0, a(n) = 2*A057627(n), otherwise, a(n) = 1 + 2 * A013928(n). [Here mu is Moebius mu-function, A008683, which is zero only when n is a nonsquarefree number, one of the numbers in A013929].

For all n, A000035(a(n)) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) modulo 2.

cp's OEIS Frontend

A072061 [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], ..., where t=tau = (1+sqrt(5))/2 and []=floor.

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