A002689 -id:A002689 - cp's OEIS frontend

A169971 Erroneous version of A002689.

1, 6, 8, 180, 32, 10080, 3456, 453600, 115200, 47900160, 71680, 217945728000, 36578304000, 2241727488000, 45984153600, 2000741783040000

Offset: 0

dead

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A093048 a(n) = n minus exponent of 2 in n, with a(0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 5, 7, 5, 9, 9, 11, 10, 13, 13, 15, 12, 17, 17, 19, 18, 21, 21, 23, 21, 25, 25, 27, 26, 29, 29, 31, 27, 33, 33, 35, 34, 37, 37, 39, 37, 41, 41, 43, 42, 45, 45, 47, 44, 49, 49, 51, 50, 53, 53, 55, 53, 57, 57, 59, 58, 61, 61, 63, 58, 65, 65, 67, 66, 69

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

			G.f. = x + x^2 + 3*x^3 + 2*x^4 +  5*x^5 + 5*x^6 + 7*x^7 + 5*x^8 + 9*x^9 + ... - _Michael Somos_, Jan 25 2020

R. J. Mathar, Table of n, a(n) for n = 0..1000

a(n) = n - A007814(n) = A093049(n) + 1, n > 0.
a(n) is the exponent of 2 in A002689(n-1), A014070(n), A060690(n), A075101(n).
See also A084623.

A093048 := proc(n)
    n-A007814(n) ;
end proc: # R. J. Mathar, Jul 24 2014

a[ n_] := If[ n == 0, n - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

a(n) = if(n<1, 0, if(n%2==0, a(n/2) + n/2 - 1, n))

a(n) = n - valuation(n, 2) \\ Jianing Song, Oct 24 2018

def A093048(n): return n-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n + 1.
G.f.: Sum_{k>=0} (t*(t^3 + t^2 + 1)/(1 - t^2)^2), with t = x^2^k.
a(n) = Sum_{k=1..n} sign(n mod 2^k). - Wesley Ivan Hurt, May 09 2021

A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

Original entry on oeis.org

1, 1, 7, 17, 41, 731, 8563, 27719, 190073, 516149, 1013143139, 1519024289, 14108351869, 14399405173, 23142912688967, 83945247395407, 84894728616107, 3204549982389941, 262488267575333123, 9027726081126601799, 2026692221793223022131, 1375035304877251309001

Offset: 0

Paul Curtz, Aug 27 2011

Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).
Reciprocally
1,   1/2,   5/12,     3/8, 251/720,    95/288, 19087/60480, 5257/17280,
1/2, 1/6,    1/8,  19/180,    3/32, 863/10080,    275/3456,
1/3, 1/12, 7/120,  17/360, 41/1008, 731/20160, 8563/259200,
1/4, 1/20, 1/30,   11/420, 89/4032,5849/302400,
1/5, 1/30, 3/140, 83/5040, 59/4320,
1/6, 1/42, 5/336,
1/7, 1/56,
1/8.
First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.
Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).
Third row: a(n)/A194506(n).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.

Cf. A194506 (denominator).

read("transforms3") ;
L := [seq(1/n,n=1..20)] ;
L1 := AKIYAMATANIGAWAi(L) ;
L2 := AKIYATANI(L1) ;
L3 := AKIYATANI(L2) ;
apply(numer,%) ; # R. J. Mathar, Aug 27 2011
# second Maple program:
b:= proc (n, k) option remember;
      `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
    end:
a:= n-> numer(b(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 27 2011

a[n_, 0] := 1/(n+1); a[n_, m_] := a[n, m] = a[n, m-1] - a[n+1, m-1]/m; Table[a[2, m], {m, 0, 21}] // Numerator (* Jean-François Alcover, Aug 09 2012 *)
Numerator@Table[(-1)^n (n + 1) Integrate[FunctionExpand[x Binomial[x, n + 1]], {x, 0, 1}], {n, 0, 20}] (* Vladimir Reshetnikov, Feb 01 2017 *)

a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{0Vladimir Reshetnikov, Feb 01 2017

A174727 a(n) = A091137(n+1)/(n+1).

Original entry on oeis.org

2, 6, 8, 180, 288, 10080, 17280, 453600, 806400, 47900160, 87091200, 217945728000, 402361344000, 2241727488000, 4184557977600, 2000741783040000, 3766102179840000, 2838385676206080000, 5377993912811520000, 1686001091666411520000, 3211430650793164800000, 423033001181754163200000

Offset: 0

Paul Curtz, Mar 28 2010

nonn

Previous name: Inverse Akiyama-Tanigawa algorithm. From a column instead of a row. Bernoulli case A164555/A027642. We start from column 1, 1/2, 1/3, 1/4, 1/5 = A000012/A000027. First row: 1) (unreduced) 1, 1/2, 5/12, 9/24, 251/720 = A002657/A091137 (Cauchy from Bernoulli) (*); 2) (reduced) 1, 1/2, 5/12, 3/8, 251/720 = A002208/A002209 (Stirling and Bernoulli). Unreduced second row: 1/2, 1/6, 1/8, 19/180, 27/288, 863/10080 = A141417(n+1)/a(n).

(*) Reference page 56 (first row) and page 36 (upper main diagonal). From J. C. Adams (and Bashforth) numerical integration. See A165313 and A147998. See A002206 logarithm numbers (Gregory).

P. Curtz, Intégration numérique des systèmes différentiels .. . Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.

Cf. A002689, A091137.

A091137[n_] := A091137[n] = Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]*A091137[n-1]; A091137[0] = 1; a[n_] := A091137[n+1]/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover_, Aug 14 2012 *)

f(n) = my(r =1); forprime(p=2, n+1, r*=p^(n\(p-1))); r; \\ A091137
a(n) = f(n+1)/(n+1); \\ Michel Marcus, Jun 30 2019

a(n) = A091137(n+1)/(n+1).

Extended up to a(18) by Jean-François Alcover, Aug 14 2012

New name and more terms from Michel Marcus, Jun 30 2019

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A169971 Erroneous version of A002689.

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A093048 a(n) = n minus exponent of 2 in n, with a(0) = 0.

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A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

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A174727 a(n) = A091137(n+1)/(n+1).

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