A242094 -id:A242094 - cp's OEIS frontend

A003256 a(n) is the number m such that A242094(m) = A001950(n).

2, 5, 7, 9, 12, 14, 17, 19, 21, 24, 26, 28, 31, 33, 36, 38, 40, 43, 45, 47, 49, 51, 54, 56, 58, 61, 63, 66, 68, 70, 73, 75, 77, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 116, 118, 120, 123, 125, 127, 129, 131, 134, 136, 138, 141, 143

Offset: 1

N. J. A. Sloane

nonn

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

Ron Reble remarks that Carlitz has a typo on page 339: Carlitz writes "In particular since (b) is a proper subset of (a), there exists a function v such that b = av." It should be "(b) is a proper subset of (u), ... b = uv." - N. J. A. Sloane, Jan 20 2020

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

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

Cf. A001950, A003234, A242094.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a003256 = (+ 1) . fromJust . (`elemIndex` a242094_list) . a001950
-- Reinhard Zumkeller, Oct 03 2014

a(n) = A001950(n) - j, where j is the largest integer such that A003234(j) < n. [Carlitz, Thm. 7.3]. - Eric M. Schmidt, Sep 16 2014

New definition by Eric M. Schmidt, Aug 17 2014

A247421 The number m such that A242094(m) = A005206(A003234(n)).

Original entry on oeis.org

2, 5, 7, 9, 11, 12, 14, 17, 19, 21, 24, 26, 28, 30, 31, 33, 36, 38, 40, 42, 43, 45, 47, 49, 51, 54, 56, 58, 60, 61, 63, 66, 68, 70, 72, 73, 75, 77, 79, 80, 82, 85, 87, 89, 91, 92, 94, 97, 99, 101, 104, 106, 108, 110, 111, 113, 116, 118, 120, 122, 123, 125, 127

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named x in [Carlitz].

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

A003249 a(n) = A001950(A003234(n)) + 1.

Original entry on oeis.org

8, 21, 29, 42, 50, 55, 63, 76, 84, 97, 110, 118, 131, 139, 144, 152, 165, 173, 186, 194, 199, 207, 220, 228, 241, 254, 262, 275, 283, 288, 296, 309, 317, 330, 338, 343, 351, 364, 372, 377, 385, 398, 406, 419, 427, 432, 440, 453, 461, 474, 487, 495, 508, 516

Offset: 1

N. J. A. Sloane

nonn

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

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

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

Cf. A242094 (complement), A001950, A003234.

a003249 = (+ 1) . a001950 . a003234 -- Reinhard Zumkeller, Oct 03 2014

A003249 := proc(n)
    A001950(A003234(n)) +1 ;
end proc:
seq(A003249(n),n=1..80) ; # R. J. Mathar, Jul 16 2024

nmax = 80;
A001950[n_] := Floor[n*GoldenRatio^2];
A003231[n_] := 2*n + Floor[n*GoldenRatio];
A003234 = Select[Range[4*nmax],
   A003231[A001950[#]] == A001950[A003231[#]] - 1&];
a[n_] := A001950[A003234[[n]]] + 1;
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Jul 21 2024 *)

Corrected and extended by, and a definition from Eric M. Schmidt, Aug 14 2014

A247419 a(2n) = A003256(n); a(2n-1) = A003256(n) - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named w in [Carlitz] (cf. Thm. 7.9), which defines this sequence by the property A242094(a(n)) is the n-th positive integer not of the form A000201(A001950(m)).

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

import Data.List (transpose)
a247419 n = a247419_list !! (n-1)
a247419_list = concat $
               transpose [map (subtract 1) a003256_list, a003256_list]
-- Reinhard Zumkeller, Oct 03 2014

A247423 Complement of A247424.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 51, 52, 54, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 103, 104, 106, 108, 110

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named y in [Carlitz], which defines this sequence by the property A242094(A247419(a(n))) = A005206(A003234(n)).

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

A247420 Complement of A247419.

Original entry on oeis.org

3, 10, 15, 22, 29, 34, 41, 52, 59, 64, 71, 78, 83, 90, 95, 102, 109, 114, 121, 132, 139, 144, 151, 158, 163, 170, 181, 188, 193, 200, 207, 212, 219, 224, 231, 238, 243, 250, 261, 268, 273, 280, 287, 292, 299, 310, 317, 322, 329, 340, 347, 352, 359, 366, 371

Offset: 1

Eric M. Schmidt, Sep 16 2014

nonn

This is the function named w' in [Carlitz].

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

a(n) = 2*A000201(A000201(A242094(n))) + n. [Carlitz, Cor. 7.12]

cp's OEIS Frontend

A003256 a(n) is the number m such that A242094(m) = A001950(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Formula

Extensions

A247421 The number m such that A242094(m) = A005206(A003234(n)).

Views

Author

Keywords

Comments

Links

A003249 a(n) = A001950(A003234(n)) + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A247419 a(2n) = A003256(n); a(2n-1) = A003256(n) - 1.

Views

Author

Keywords

Comments

Links

Programs

Haskell

A247423 Complement of A247424.

Views

Author

Keywords

Comments

Links

A247420 Complement of A247419.

Views

Author

Keywords

Comments

Links

Formula