A003233 Numbers k such that A003231(A001950(k)) = A001950(A003231(k)).

1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

See 3.3 p. 344 in Carlitz link. - Michel Marcus, Feb 02 2014

This is the function named r in [Carlitz]. - Eric M. Schmidt, Aug 14 2014

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

Crossrefs

Cf. A001950, A003231, A003234.

Programs

Haskell

a003233 n = a003233_list !! (n-1)
a003233_list = [x | x <- [1..],
                    a003231 (a001950 x) == a001950 (a003231 x)]
-- Reinhard Zumkeller, Oct 03 2014

Mathematica

a3221[n_] := Floor[n(5 + Sqrt[5])/2];
a1950[n_] := Floor[n(1 + Sqrt[5])^2/4];
Select[Range[100], a3221[a1950[#]] == a1950[a3221[#]]&] (* Jean-François Alcover, Aug 04 2018 *)

PARI

A001950(n) = floor(n*(sqrt(5)+3)/2);
A003231(n) = floor(n*(sqrt(5)+5)/2);
lista(nn) = { for(n=1, nn, if (A003231(A001950(n)) == A001950(A003231(n)), print1(n, ", ")));} \\ Michel Marcus, Feb 02 2014

Python

from math import isqrt
from itertools import count, islice
def A003233_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:((m:=(n+isqrt(5*n**2)>>1)+n)+isqrt(5*m**2)>>1)+(m<<1)==((k:=(n+isqrt(5*n**2)>>1)+(n<<1))+isqrt(5*k**2)>>1)+k,count(max(1,startvalue)))
A003233_list = list(islice(A003233_gen(),30)) # Chai Wah Wu, Sep 02 2022

Extensions

More terms from Michel Marcus, Feb 02 2014

Definition from Michel Marcus moved from comment to name by Eric M. Schmidt, Aug 17 2014