A106619 -id:A106619 - cp's OEIS frontend

A026741 a(n) = n if n odd, n/2 if n even.

0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34, 69, 35, 71, 36, 73, 37, 75, 38

Offset: 0

J. Carl Bellinger (carlb(AT)ctron.com)

a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002
a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington, Oct 23 2003
For n > 1, a(n) is the greatest common divisor of all permutations of {0, 1, ..., n} treated as base n + 1 integers. - David Scambler, Nov 08 2006 (see the Mathematics Stack Exchange link below).
From Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009: (Start)
Sequence A075888 and the above sequence are fitting together.
First 2 entries of this sequence have to be taken out.
In some cases two three or more sequenced entries of this sequence have to be added together to get the next entry of A075888.
Example: Sequences begin with 1, 3, 2, 5, 3, 7, 4, 9 (4 + 9 = 13, the next entry in A075888).
But it works out well up to primes around 50000 (haven't tested higher ones).
As A075888 gives a very regular graph. There seems to be a regularity in the primes. (End)
Starting with 1 = triangle A115359 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009
From Gary W. Adamson, Dec 11 2009: (Start)
Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column, shifted down twice. This sequence starting with 1 = M * (1, 2, 3, ...)
M =
  1;
  1, 0;
  1, 1, 0;
  0, 1, 0, 0;
  0, 1, 1, 0, 0;
  0, 0, 1, 0, 0, 0;
  0, 0, 1, 1, 0, 0, 0;
  ...
A026741 = M * (1, 2, 3, ...); but A002487 = lim_{n->infinity} M^n, a left-shifted vector considered as a sequence. (End)
A particular case of sequence for which a(n+3) = (a(n+2) * a(n+1)+q)/a(n) for every n > n0. Here n0 = 1 and q = -1. - Richard Choulet, Mar 01 2010
For n >= 2, a(n+1) is the smallest m such that s_n(2*m*(n-1))/(n-1) is even, where s_b(c) is the sum of digits of c in base b. - Vladimir Shevelev, May 02 2011
A001477 and A005408 interleaved. - Omar E. Pol, Aug 22 2011
Numerator of n/((n-1)*(n-2)). - Michael B. Porter, Mar 18 2012
Number of odd terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013
For n >= 3, a(n) is the periodic of integer of spiral length ratio of spiral that have (n-1) circle centers. See illustration in links. - Kival Ngaokrajang, Dec 28 2013
This is the sequence of Lehmer numbers u_n(sqrt(R), Q) with the parameters R = 4 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all natural numbers n and m. Cf. A005013 and A108412. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) = 2 begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [0, 1, 4, 3, 8, 5, 12, ...] is A022998, also a strong divisibility sequence. - Peter Bala, May 19 2014
For n >= 3, (a(n-2)/a(n))*Pi = vertex angle of a regular n-gon. See illustration in links. - Kival Ngaokrajang, Jul 17 2014
For n > 1, the numerator of the harmonic mean of the first n triangular numbers. - Colin Barker, Nov 13 2014
The difference sequence is a permutation of the integers. - Clark Kimberling, Apr 19 2015
From Timothy Hopper, Feb 26 2017: (Start)
Given the function a(n, p) = n/p if n mod p = 0, else n, then a possible formula is: a(n, p) = n*(1 + (p-1)*((n^(p-1)) mod p))/p, p prime, (n^(p-1)) mod p = 1, n not divisible by p. (Fermat's Little Theorem). Examples: p = 2; a(n), p = 3; A051176(n), p = 5; A060791(n), p = 7; A106608(n).
Conjecture: lcm(n, p) = p*a(n, p), gcd(n, p) = n/a(n, p). (End)
Let r(n) = (a(n+1) + 1)/a(n+1) if n mod 2 = 1, a(n+1)/(a(n+1) + 2) otherwise; then lim_{k->oo} 2^(k+2) * Product_{n=0..k} r(n)^(k-n) = Pi. - Dimitris Valianatos, Mar 22 2021
Number of integers k from 1 to n such that gcd(n,k) is odd. - Amiram Eldar, May 18 2025

			G.f. = x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 3*x^6 + 7*x^7 + 4*x^8 + ...

David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 53.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. Penguin (1997), p. 79.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Y. Ito and I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.
Mathematics Stack Exchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.
Kival Ngaokrajang, Illustration of spiral with circle centers 2..5.
Kival Ngaokrajang, Illustration of vertex angle of regular n-gon for n = 3..7.
Eric Weisstein's World of Mathematics, Lehmer Number.
Eric Weisstein's World of Mathematics, Simplex Simplex Picking.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Index to divisibility sequences.

Signed version is in A030640. Partial sums give A001318.
Cf. A051176, A060819, A060791, A060789 for n / gcd(n, k) with k = 3..6. See also A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).
Cf. A045896, A022998, A060762, A126988, A109007, A130334, A109043, A115359, A002487, A220466.
Cf. A013942.
Cf. A227042 (first column). Cf. A005013 and A108412.
Cf. A000217, A256095.

import Data.List (transpose)
a026741 n = a026741_list !! n
a026741_list = concat $ transpose [[0..], [1,3..]]
-- Reinhard Zumkeller, Dec 12 2011

[2*n/(3+(-1)^n): n in [0..70]]; // Vincenzo Librandi, Aug 14 2011

A026741 := proc(n) if type(n,'odd') then n; else n/2; end if; end proc: seq(A026741(n), n=0..76); # R. J. Mathar, Jan 22 2011

Numerator[Abs[Table[Det[DiagonalMatrix[Table[1/i^2 - 1, {i, 1, n - 1}]] + 1], {n, 20}]]] (* Alexander Adamchuk, Jun 02 2006 *)
halfMax = 40; Riffle[Range[0, halfMax], Range[1, 2halfMax + 1, 2]] (* Harvey P. Dale, Mar 27 2011 *)
a[ n_] := Numerator[n / 2]; (* Michael Somos, Jan 20 2017 *)
Array[If[EvenQ[#],#/2,#]&,80,0] (* Harvey P. Dale, Jul 08 2023 *)

a(n) = numerator(n/2) \\ Rick L. Shepherd, Sep 12 2007

def A026741(n): return n if n % 2 else n//2 # Chai Wah Wu, Apr 02 2021

[lcm(n, 2) / 2 for n in range(77)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 + x + x^2)/(1-x^2)^2. - Len Smiley, Apr 30 2001
a(n) = 2*a(n-2) - a*(n-4) for n >= 4.
a(n) = n * 2^((n mod 2) - 1). - Reinhard Zumkeller, Oct 16 2001
a(n) = 2*n/(3 + (-1)^n). - Benoit Cloitre, Mar 24 2002
Multiplicative with a(2^e) = 2^(e-1) and a(p^e) = p^e, p > 2. - Vladeta Jovovic, Apr 05 2002
a(n) = n / gcd(n, 2). a(n)/A045896(n) = n/((n+1)*(n+2)).
For n > 0, a(n) = denominator of Sum_{i=1..n-1} 2/(i*(i+1)), numerator=A022998. - Reinhard Zumkeller, Apr 21 2012, Jul 25 2002 [thanks to Phil Carmody who noticed an error]
For n > 1, a(n) = GCD of the n-th and (n-1)-th triangular numbers (A000217). - Ross La Haye, Sep 13 2003
Euler transform of finite sequence [1, 2, -1]. - Michael Somos, Jun 15 2005
G.f.: x * (1 - x^3) / ((1 - x) * (1 - x^2)^2) = Sum_{k>0} k * (x^k - x^(2*k)). - Michael Somos, Jun 15 2005
a(n+3) + a(n+2) = 3 + a(n+1) + a(n). a(n+3) * a(n) = - 1 + a(n+2) * a(n+1). a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 15 2005
For n > 1, a(n) is the numerator of the average of 1, 2, ..., n - 1; i.e., numerator of A000217(n-1)/(n-1), with corresponding denominators [1, 2, 1, 2, ...] (A000034). - Rick L. Shepherd, Jun 05 2006
Equals A126988 * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
For n >= 1, a(n) = gcd(n,A000217(n)). - Rick L. Shepherd, Sep 12 2007
a(n) = numerator(n/(2*n-2)) for n >= 2; A022998(n-1) = denominator(n/(2*n-2)) for n >= 2. - Johannes W. Meijer, Jun 18 2009
a(n) = A167192(n+2, 2). - Reinhard Zumkeller, Oct 30 2009
a(n) = A106619(n) * A109012(n). - Paul Curtz, Apr 04 2011
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109043(n)/2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s). (End)
a(n) = A001318(n) - A001318(n-1) for n > 0. - Jonathan Sondow, Jan 28 2013
a((2*n+1)*2^p - 1) = 2^p - 1 + n*A151821(p+1), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 03 2013
a(n+1) = denominator(H(n, 1)), n >= 0, with H(n, 1) = 2*n/(n+1) the harmonic mean of n and 1. a(n+1) = A227042(n, 1). See the formula a(n) = n/gcd(n, 2) given above. - Wolfdieter Lang, Jul 04 2013
a(n) = numerator(n/2). - Wesley Ivan Hurt, Oct 02 2013
a(n) = numerator(1 - 2/(n+2)), n >= 0; a(n) = denominator(1 - 2/n), n >= 1. - Kival Ngaokrajang, Jul 17 2014
a(n) = Sum_{i = floor(n/2)..floor((n+1)/2)} i. - Wesley Ivan Hurt, Apr 27 2016
Euler transform of length 3 sequence [1, 2, -1]. - Michael Somos, Jan 20 2017
G.f.: x / (1 - x / (1 - 2*x / (1 + 7*x / (2 - 9*x / (7 - 4*x / (3 - 7*x / (2 + 3*x))))))). - Michael Somos, Jan 20 2017
From Peter Bala, Mar 24 2019: (Start)
a(n) = Sum_{d|n, n/d odd} phi(d), where phi(n) is the Euler totient function A000010.
O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 - x^(2*n)). (End)
a(n) = A256095(2*n,n). - Alois P. Heinz, Jan 21 2020
E.g.f.: x*(2*cosh(x) + sinh(x))/2. - Stefano Spezia, Apr 28 2023
From Ctibor O. Zizka, Oct 05 2023: (Start)
For k >= 0, a(k) = gcd(k + 1, k*(k + 1)/2).
If (k mod 4) = 0 or 2 then a(k) = (k + 1).
If (k mod 4) = 1 or 3 then a(k) = (k + 1)/2. (End)
Sum_{n=1..oo} 1/a(n)^2 = 7*Pi^2/24. - Stefano Spezia, Dec 02 2023
a(n)*a(n+1) = A000217(n). - Rémy Sigrist, Mar 19 2025

Better description from Jud McCranie

Edited by Ralf Stephan, Jun 04 2003

A106617 Numerator of n/(n+16).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 4, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 75, 19, 77, 39, 79

Offset: 0

N. J. A. Sloane, May 15 2005

A multiplicative sequence. Also, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

			From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - 2*G(x^2) - 4*G(x^4) - 8*G(x^8) - 16*G(x^16), where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} n^2*a(n)*x^n = H(x) - 2^2*H(x^2) - 4^2*H(x^4) - 8^2*H(x^8) - 16^2*H(x^16), where H(x) = x*(1 + 4*x + x^2)/(1 - x)^4. In general, the o.g.f. for Sum_{n >= 1} (n^k*a(n))*x^n for positive k involves the Eulerian polynomials.
In the other direction,
Sum_{n >= 1} (a(n)/n)*x^n = J(x) - (1/2)*J(x^2) - (1/4)*J(x^4) - (1/8)*J(x^8) - (1/16)*J(x^16), where J(x) = x/(1 - x).
Sum_{n >= 1} (a(n)/n^2)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4^2)*L(x^4) - (1/8^2)*L(x^8) - (1/16^2)*L(x^16), where L(x) = log(1/(1 - x)). In general, the o.g.f. for Sum_{n >= 0} (a(n)/n^k)*x^n, for k >= 3, involves the polylogarithm Li_(k-1)(x).
Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4) + (1/2)*L(x^8) + (1/2)*L(x^16). (End)

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A106614, A106615, A106616, A106618, A106619, A106620, A106621.

Cf. Other sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([0..100],n->NumeratorRat(n/(n+16))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+16)): n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

seq(numer(n/(n+16)), n=0..100); # Nathaniel Johnston, Apr 18 2011

Numerator/@Table[n/(n+16),{n,0,10}] (* Harvey P. Dale, Apr 26 2011 *)

a(n)=n>>min(valuation(n,2),4) \\ Charles R Greathouse IV, Jun 17 2012

[lcm(n,16)/16 for n in range(0, 100)] # Zerinvary Lajos, Jun 12 2009

a(n) = 2*a(n-16) - a(n-32) for n > 31. - Paul Curtz, Apr 12 2011
Octosections: a(8*n) = A026741(n). a(2+8*n) = 1+4*n. a(4+8*n) = 1+2*n. a(6+8*n) = 3+4*n. Bisection: a(1+2*n) = 1+2*n. - Paul Curtz, Apr 12 2011
Dirichlet g.f.: zeta(s-1)*(1-1/2^s-1/4^s-1/8^s-1/16^s). - R. J. Mathar, Apr 18 2011
a(n) = numerator of n/(2^(2*n+1)). - Ralf Steiner, Feb 09 2017
The previous comment is incorrect, a(n) first differs from the numerator of n/(2^(2*n+1)) at n = 32. - Peter Bala, Feb 27 2019
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,16), where gcd(n,16) = [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, ...] is a periodic sequence of period 16: a(n) is thus quasi_polynomial in n.
O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4) - F(x^8) - F(x^16), where F(x) = x/(1 - x)^2.
More generally, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) + F(m,x^8) + F(m,x^16) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m-th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences ( (n^m)*a(n) )n>=1 for m in Z. Some examples are given below. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^max(0,e-4), and a(p^e) = p^e if p>2.
Sum_{k=1..n} a(k) ~ (171/512) * n^2. (End)

A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

Original entry on oeis.org

1, 3, 1, 2, 5, 1, 5, 3, 7, 1, 3, 7, 4, 9, 1, 7, 1, 1, 5, 11, 1, 4, 9, 5, 11, 6, 13, 1, 9, 5, 11, 3, 13, 7, 15, 1, 5, 11, 2, 13, 7, 5, 8, 17, 1, 11, 3, 13, 7, 3, 2, 17, 9, 19, 1, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 1

Offset: 1

Wolfdieter Lang, Jul 01 2013

See the comments under A227041. a(n,m) gives the denominator of H(n,m) = 2*n*m/(n+m) in lowest terms.

			The triangle of denominators of H(n,m), called a(n,m) begins:
n\m  1   2   3   4   5    6    7    8    9   10  11 ...
1:   1
2:   3   1
3:   2   5   1
4:   5   3   7   1
5:   3   7   4   9   1
6:   7   1   1   5  11    1
7:   4   9   5  11   6   13    1
8;   9   5  11   3  13    7   15    1
9:   5  11   2  13   7    5    8   17    1
10: 11   3  13   7   3    2   17    9   19    1
11:  6  13   7  15   8   17    9   19   10   21   1
...
For the triangle of the rationals H(n,m) see the example section of A227041.
H(4,2) = denominator(16/6) = denominator(8/3) = 3 = 6/gcd(6,8) = 6/2.

Eric Weisstein's World of Mathematics, Harmonic Mean.

Cf. A227041, A026741 (column m=1), A000265 (m=2), A106619 (m=3), A227140(n+8) (m=4), A227108 (m=5), A221918/A221919.

a(n,m) = denominator(2*n*m/(n+m)), 1 <= m <= n.

a(n,m) = (n+m)/gcd(2*n*m, n+m) = (n+m)/gcd(n+m, 2*m^2), 1 <= m <= n.

A106616 Numerator of n/(n+15).

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 2, 7, 8, 3, 2, 11, 4, 13, 14, 1, 16, 17, 6, 19, 4, 7, 22, 23, 8, 5, 26, 9, 28, 29, 2, 31, 32, 11, 34, 7, 12, 37, 38, 13, 8, 41, 14, 43, 44, 3, 46, 47, 16, 49, 10, 17, 52, 53, 18, 11, 56, 19, 58, 59, 4, 61, 62, 21, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26

Offset: 0

N. J. A. Sloane, May 15 2005

Multiplicative and also a strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A000010, A106614, A106615, A106617, A106618, A106619, A106620, A106621.

List([0..80],n->NumeratorRat(n/(n+15))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+15)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+15)),n=0..100); # Nathaniel Johnston, Apr 18 2011

f[n_]:=Numerator[n/(n+15)]; Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011*)

a(n) = numerator(n/(n+15)); \\ Michel Marcus, Feb 19 2017

[lcm(n,15)/15 for n in range(0, 79)] # Zerinvary Lajos, Jun 09 2009

Dirichlet g.f.: zeta(s-1)*(1-4/5^s-2/3^s+8/15^s). - R. J. Mathar, Apr 18 2011
a(n) = gcd((n-2)*(n-1)*n*(n+1)*(n+2)/15, n) for n>=1. - Lechoslaw Ratajczak, Feb 19 2017
From Peter Bala, Feb 24 2019: (Start)
a(n) = n/gcd(n,15), a quasi-polynomial in n since gcd(n,15) is a purely periodic sequence of period 15.
O.g.f.: F(x) - 2*F(x^3) - 4*F(x^5) + 8*F(x^15), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 15} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (2/3)*log(1/(1 - x^3)) + (4/5)*log(1/(1 - x^5)) + (8/15)*log(1/(1 - x^15)), where phi(n) denotes the Euler totient function A000010. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(3^e) = 3^max(0,e-1), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.
Sum_{k=1..n} a(k) ~ (49/150) * n^2. (End)

A145911 a(n) = A145909(n)/8.

Original entry on oeis.org

0, 1, 1, 2, 10, 5, 7, 28, 4, 5, 55, 22, 26, 91, 35, 40, 136, 17, 19, 190, 70, 77, 253, 92, 100, 325, 39, 42, 406, 145, 155, 496, 176, 187, 595, 70, 74, 703, 247, 260, 820, 287, 301, 946, 110, 115, 1081, 376, 392, 1225, 425, 442, 1378

Offset: 0

Paul Curtz, Oct 24 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,1).

Cf. A061039, A106619, A145909.

A061039:= proc(n) numer(1/9-1/n^2) ; end proc:
A145909:= proc(n) ; A061039(6*n+3); end proc:
A145911:= proc(n) ; A145909(n)/8; end proc:
seq(A145911(n),n=0..80) ; # R. J. Mathar, Jan 06 2011

Numerator[(1/9)*(1 -1/(2*Range[0, 75] +1)^2)]/8 (* G. C. Greubel, Mar 12 2022 *)

[numerator((1/9)*(1 - 1/(2*n+1)^2))/8 for n in (0..75)] # G. C. Greubel, Mar 12 2022

a(n) = A106619(n)*A106619(n+1).

A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, 3, 1, -1, -3, -1, 1, 2, 1, 0, -1, -2, -1, 1, 5, 3, 1, -1, -3, -5, -1, 1, 3, 1, 1, 0, -1, -1, -3, -1, 1, 7, 5, 1, 1, -1, -1, -5, -7, -1, 1, 4, 3, 2, 1, 0, -1, -2, -3, -4, -1, 1, 0, 7, 5, 3, 1, -1, -3, -5, -7, -9, -1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367727.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the numerators begins:
  1, -1, -1, -1, -1, -1, -1, -1, ...
  1,  0, -1, -1, -3, -2, -5, -3, ...
  1,  1,  0, -1, -1, -3, -1, -5, ...
  1,  1,  1,  0, -1, -1, -1, -2, ...
  1,  3,  1,  1,  0, -1, -1, -3, ...
  1,  2,  3,  1,  1,  0, -1, -1, ...
  1,  5,  1,  1,  1,  1,  0, -1, ...
  1,  3,  5,  2,  3,  1,  1,  0, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367825 (denominator), A367826 (antidiagonal sums).

Cf. A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106617, A106619, A153881 (n=0), A231190, A367727 (k=1).

A[0,0]=1; A[n_,k_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(1, n) = -A026741(n-1) for n > 0.
A(2, n) = -A060819(n-2) for n > 2.
A(3, n) = -A060789(n-3) for n > 3.
A(4, n) = -A106609(n-4) for n > 3.
A(5, n) = -A106611(n-5) for n > 4.
A(6, n) = -A051724(n-6) for n > 5.
A(7, n) = -A106615(n-7) for n > 6.
A(8, n) = -A106617(n-8) = A231190(n) for n > 7.
A(9, n) = -A106619(n-9) for n > 8.
A(10, n) = -A106612(n-10) for n > 9.

A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 3, 1, 3, 3, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 4, 2, 4, 1, 4, 2, 4, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 4, 6, 12, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367728.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the denominators begins:
  1, 1, 1, 1,  1,  1,  1,  1, ...
  1, 1, 3, 2,  5,  3,  7,  4, ...
  1, 3, 1, 5,  3,  7,  2,  9, ...
  1, 2, 5, 1,  7,  4,  3,  5, ...
  1, 5, 3, 7,  1,  9,  5, 11, ...
  1, 3, 7, 4,  9,  1, 11,  6, ...
  1, 7, 2, 3,  5, 11,  1, 13, ...
  1, 4, 9, 5, 11,  6, 13,  1, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367824 (numerator), A367827 (antidiagonal sums).

Cf. A000012 (n=0), A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106615, A106617, A231190, A367728 (k=1).

A[0,0]=1; A[n_,k_]:=Denominator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(1, n) = A026741(n+1).
A(2, n) = A060819(n+2).
A(3, n) = A060789(n+3).
A(4, n) = A106609(n+4).
A(5, n) = A106611(n+5).
A(6, n) = A051724(n+6).
A(7, n) = A106615(n+7).
A(8, n) = A106617(n+8) = A231190(n+16).
A(9, n) = A106619(n+9).
A(10, n) = A106612(n+10).

A185727 Integers of the form A145911(k)/(k+1) sorted along increasing k.

Original entry on oeis.org

0, 2, 1, 5, 2, 8, 1, 11, 4, 14, 5, 17, 2, 20, 7, 23, 8, 26, 3, 29, 10, 32, 11, 35, 4, 38, 13, 41, 14, 44, 5, 47, 16, 50, 17, 53, 6, 56, 19, 59, 20, 62, 7, 65, 22, 68, 23, 71, 8, 74, 25, 77, 26, 80, 9, 83, 28, 86, 29, 89, 10, 92, 31, 95, 32, 98, 11, 101, 34, 104, 35, 107, 12, 110, 37, 113, 38, 116, 13, 119, 40, 122

Offset: 0

Paul Curtz, Feb 05 2011

The standard offset is changed to zero to simplify formulas related to the a(n).
The sequence of fractions A145911(k)/(k+1) is 0,  1/2,  1/3, 1/2,  2,  5/6, 1,  7/2,  4/9, 1/2,  5, 11/6, 2, 13/2,  7/3,  5/2,  8, 17/18, 1, 19/2, 10/3, 7/2, 11, 23/6, 4, 25/2, 13/9, 3/2, 14, 29/6, ....
Its numerators are A106619. Integer values appear at indices of the form 6*n and 4+6*n.
The sequence of denominators of the fractions appears to have a period of length 18.
a(n+18)-a(n) = 3*(a(n+6)-a(n)) = 3, 27, 9, 27, 9, 27, 3, 27, 9, ,... are multiples of 3, apparently with a period of length 6.
The recurrence a(n) = 2a(n-6)-a(n-12) shows that the sequence consists of 6 interleaved first-order polynomials: a(6*n)=n. a(1+6*n) = 2+9*n. a(2+6*n) = 1+3*n = A016777(n). a(3+6*n) = 5+9*n. a(4+6*n) = 2+3*n = A016789(n). a(5+6*n) = 8+9*n. - Paul Curtz, Feb 23 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000

A106619 := proc(n) numer(n/(n+18)) ; end proc:
A185727 := proc(n) if type(n,'even') then A106619(3*n) ; else A106619(3*n+1) ; end if; end proc:
seq(A185727(n),n=0..80) ; # R. J. Mathar, Feb 18 2011

CoefficientList[Series[x*(2 + x + 5*x^2 + 2*x^3 + 8*x^4 + x^5 + 7*x^6 + 2*x^7 + 4*x^8 + x^9 + x^10)/((x - 1)^2*(1 + x)^2*(1 + x + x^2)^2*(x^2 - x + 1)^2), {x,0,50}], x] (* G. C. Greubel, Jul 11 2017 *)

x='x+O('x^50); concat([0], Vec(x*(2 + x + 5*x^2 + 2*x^3 + 8*x^4 + x^5 + 7*x^6 + 2*x^7 + 4*x^8 + x^9 + x^10)/((x - 1)^2*(1 + x)^2*(1 + x + x^2)^2*(x^2 - x + 1)^2))) \\ G. C. Greubel, Jul 11 2017

a(2*n) = A051176(n) = A106619(6n).
a(1+2*n) = 2+3*n = A106619(4+6*n).
a(6*n) = n.
From R. J. Mathar, Feb 10 2011: (Start)
a(n)= +2*a(n-6) -a(n-12).
G.f.:  x*(2+x +5*x^2 +2*x^3 +8*x^4 +x^5 +7*x ^6 +2*x^7 +4*x^8 +x^9 +x^10) / ( (x-1)^2*(1+x)^2*(1+x+x^2)^2*(x^2-x+1)^2 ). (End)
a(n) = A014682(n) if n is not a multiple of 6. - Paul Curtz, Feb 23 2011

cp's OEIS Frontend

A026741 a(n) = n if n odd, n/2 if n even.

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A106617 Numerator of n/(n+16).

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A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

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A106616 Numerator of n/(n+15).

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A145911 a(n) = A145909(n)/8.

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A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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