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 1-3 of 3 results.

A276090 Left inverse of A276089: For n = sum_{i=1..} d(i)*i! (with each d(i) <= i), a(n) = sum_{j=1..} d(2j-1)*j!.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 6
Offset: 0

Views

Author

Antti Karttunen, Aug 19 2016

Keywords

Comments

This "deaerates" A276089(n) by picking only the digits from the odd positions of its factorial base representation. Of course, when computed for an arbitrary n, those digits, when "compressed" into a(n) are not necessarily valid digits in standard factorial base representation (A007623).

Examples

			For n = 311 ("22321" in factorial base representation) we pick the digits at odd positions 1, 3 and 5, thus we get a(311) = 2*3! + 3*2! + 1*1! = 19.
For n=373 ("30201"), we pick the digits from those same positions and construct a(373) = 3*3! + 2*2! + 1*1! = 23.

Links

Crossrefs

Left inverse of A276089.
For no apparent reason, the terms a(0)..a(21) are equal to the terms a(3)..a(24) of A118777.
Cf. A007623.

Formula

Other identities. For all n >= 0:
a(A276089(n)) = n.

A051022 Interpolate 0's between each pair of digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 500, 501, 502, 503, 504, 505
Offset: 0

Views

Author

Eric W. Weisstein, Dec 11 1999

Keywords

Comments

These numbers have the same decimal and negadecimal representations.
Or fixed points of decimal negadecimal conversion. - Gerald Hillier, Apr 23 2015

Examples

			a(23) = 203.
a(99) = 909.
a(100) = 10000.
a(101) = 10001.
a(111) = 10101.

Links

Crossrefs

Cf. A039723, A063010, A092908 (primes), A092909 (on primes), A338754 (*11).
In other bases: A000695, A037314, A276089.

Programs

  • Haskell
    a051022 n = if n < 10 then n else a051022 n' * 100 + r
            where (n', r) = divMod n 10
-- Reinhard Zumkeller, Apr 20 2011
(HP 49G calculator)
« "" + SREV 0 9
  FOR i i "" + DUP 0 + SREPL DROP
  NEXT SREV OBJ->
». Gerald Hillier, Apr 23 2015
  • Maple
    M:= 3: # to get a(0) to a(10^M-1)
A:= 0:
for d from 1 to M do
  A:= seq(seq(a*100+b,b=0..9),a=A);
od:
A; # Robert Israel, Apr 23 2015
  • Mathematica
    Table[FromDigits[Riffle[IntegerDigits[n],0]],{n,0,60}] (* Harvey P. Dale, Nov 17 2013 *)
ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]];
k = 0; lst = {}; While[k < 1001, If[k == ToNegaBases[k, 10], AppendTo[ lst, k]]; k++]; lst (* Robert G. Wilson v, Jun 11 2014 *)
  • PARI
    a(n) = fromdigits(digits(n),100); \\ Kevin Ryde, Nov 07 2020
  • Python
    def a(n): return int("0".join(str(n)))
print([a(n) for n in range(56)]) # Michael S. Branicky, Aug 15 2022

Formula

Sums a_i*100^e_i with 0 <= a_i < 10.
a(n) = n if n < 10, otherwise a(floor(n/10))*100 + n mod 10. - Reinhard Zumkeller, Apr 20 2011 [Corrected by Kevin Ryde, Nov 07 2020]
a(n) = A338754(n)/11. - Kritsada Moomuang, Oct 20 2019 [Corrected by Kevin Ryde, Nov 07 2020]

Extensions

More terms and more precise definition from Jorge Coveiro, Apr 15 2004 and David Wasserman, Feb 26 2008
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar
Offset fixed by Reinhard Zumkeller, Apr 20 2012

A275959 Sum of distinct terms of A002674: a(0) = 0, a(2n) = A255411(A153880(a(n))), a(2n+1) = 1+A255411(A153880(a(n))).

Original entry on oeis.org

0, 1, 12, 13, 360, 361, 372, 373, 20160, 20161, 20172, 20173, 20520, 20521, 20532, 20533, 1814400, 1814401, 1814412, 1814413, 1814760, 1814761, 1814772, 1814773, 1834560, 1834561, 1834572, 1834573, 1834920, 1834921, 1834932, 1834933, 239500800, 239500801, 239500812, 239500813, 239501160, 239501161, 239501172, 239501173, 239520960, 239520961
Offset: 0

Views

Author

Antti Karttunen, Aug 16 2016

Keywords

Comments

Fixed points of involution A225901.
This can be also viewed as a function that reinterprets base-2 representation of n in base-((2n)!/2) where the digits are multiplied with the successive terms of A002674, thus a(0) = 0.

Links

Crossrefs

Cf. A002674, A059590, A153880, A255411, A276082, A276083, A276091.
Fixed points of A225901.
Subsequence of A275956 and of A276089.

Programs

  • Python
    from sympy import factorial as f
def a007623(n, p=2): return n if n
    0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
    x=(str(a007623(n)) + '0')[::-1]
    return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a255411(a153880(a(n//2))) if n%2==0 else 1 + a255411(a153880(a((n - 1)//2)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

Formula

a(0) = 0, a(2n) = A255411(A153880(a(n))), a(2n+1) = 1+A255411(A153880(a(n))).
a(n) = A276089(A276091(n)).
Showing 1-3 of 3 results.

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