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 1-4 of 4 results.

A121664 Inverse permutation to A096114.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 11, 10, 9, 7, 8, 12, 23, 22, 21, 19, 20, 18, 13, 14, 15, 17, 16, 24, 47, 46, 45, 43, 44, 42, 37, 38, 39, 41, 40, 36, 25, 26, 27, 29, 28, 30, 35, 34, 33, 31, 32, 48, 95, 94, 93, 91, 92, 90, 85, 86, 87, 89, 88, 84, 73, 74, 75, 77, 76, 78, 83, 82, 81, 79, 80, 72
Offset: 1

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Author

Antti Karttunen, Aug 25 2006, based on Amarnath Murthy's comments on A096114

Keywords

Links

Crossrefs

Inverse: A096114.

A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890
Offset: 0

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.
Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005
4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005
Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011
The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019
The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

Examples

			Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - _Philippe Deléham_, Jun 03 2015
From _Antti Karttunen_, Apr 13 2018, after also _Philippe Deléham_'s Jun 03 2015 example: (Start)
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........8                  12......../ \........24
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       10         15       30         20       40         60      120
  7 14   21  42     28  56   84  168    35  70  105  210   140 280  420 840
  etc.
Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.
(End)

Links

Crossrefs

Subsequences: A019565 (squarefree terms), A050376 (the left edge from 2 onward), A336882 (odd terms).
Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.

Programs

  • Mathematica
    a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)
  • PARI
    up_to_e = 13; \\ Good for computing up to n = (2^13)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; \\ Antti Karttunen, Apr 12 2018

Formula

a(0)=1; a(n+2^k)=a(n)*b(k) for n < 2^k, k = 0, 1, ... where b is A050376. - Thomas Ordowski, Mar 04 2005
The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011
From Antti Karttunen, Apr 12 & 17 2018: (Start)
a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).
a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).
(End)

Extensions

Entry revised Mar 17 2005 by N. J. A. Sloane, based on comments from several people, especially David Wasserman and Thomas Ordowski

A096115 If n = (2^k)-1, a(n) = a((n+1)/2) = k, if n = 2^k, a(n) = a(n-1)+1 = k+1, otherwise a(n) = (A000523(n)+1)*a(A035327(n-1)).

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 3, 4, 12, 24, 24, 12, 8, 8, 4, 5, 20, 40, 40, 60, 120, 120, 60, 20, 15, 30, 30, 15, 10, 10, 5, 6, 30, 60, 60, 90, 180, 180, 90, 120, 360, 720, 720, 360, 240, 240, 120, 30, 24, 48, 48, 72, 144, 144, 72, 24, 18, 36, 36, 18, 12, 12, 6, 7, 42, 84, 84, 126
Offset: 1

Views

Author

Amarnath Murthy, Jun 30 2004

Keywords

Comments

A fractal sequence. For k in range [1,(2^n)-1], a(2^n + k)/a(2^n - k) = n+1. Each n > 1 occurs 2*A045778(n) times in the sequence.

Crossrefs

Permutation of A096111, i.e. a(n) = A096111(A122199(n)-1) [Note the different starting offsets]. Cf. A096113, A052330, A096114, A096116.

Programs

Extensions

Edited, extended and Scheme code added by Antti Karttunen, Aug 25 2006

A096116 a(1)=1, if n=(2^k)+1, a(n) = k+2, otherwise a(n) = 2+A000523(n-1)+a(2+A035327(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 7, 6, 5, 11, 12, 14, 9, 10, 8, 7, 6, 13, 14, 16, 15, 20, 18, 17, 11, 12, 13, 15, 10, 11, 9, 8, 7, 15, 16, 18, 17, 22, 20, 19, 18, 24, 25, 27, 22, 23, 21, 20, 13, 14, 15, 17, 16, 21, 19, 18, 12, 13, 14, 16, 11, 12, 10, 9, 8, 17, 18, 20, 19, 24, 22, 21, 20, 26
Offset: 1

Views

Author

Amarnath Murthy, Jun 30 2004

Keywords

Comments

Each n > 1 occurs A025147(n) times in the sequence.

Links

Crossrefs

Cf. A096111, A050029, A050030, A052330, A096113, A096114, A096115.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, If[BitAnd[n - 1, n - 2] == 0, Log2[n - 1] + 2, 2 + Floor[Log2[n - 1]] + a[[2 + BitXor[n - 1, 2^Ceiling[Log2[n]] - 1]]]]], {n, 2, 74}]; a (* Ivan Neretin, Jun 24 2016 *)
  • Scheme
    (define (A096116 n) (cond ((= 1 n) 1) ((pow2? (- n 1)) (+ 2 (A000523 (- n 1)))) (else (+ 2 (A000523 (- n 1)) (A096116 (+ 2 (A035327 (- n 1))))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))
;; Antti Karttunen, Aug 25 2006

Extensions

Edited and extended by Antti Karttunen, Aug 25 2006
Showing 1-4 of 4 results.

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

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