A106617 -id:A106617 - cp's OEIS frontend

A247004 Denominator of (n+4)/gcd(n, 4)^2, a 16-periodic sequence that associates A061037 with A106617.

4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 09 2014

This sequence may also be defined as the denominators of A061037(n+3)/(n+1), or also as A060819 / A109008.

One can notice that the analog numerators [numerators of (n+4)/gcd(n, 4)^2] are A106617 left-shifted 4 places.

			Fractions begin:
1/4,  5,  3/2,  7, 1/2,  9,  5/2, 11, 3/4, 13,  7/2, 15, 1, 17,  9/2, 19,
5/4, 21, 11/2, 23, 3/2, 25, 13/2, 27, 7/4, 29, 15/2, 31, 2, 33, 17/2, 35,
...
Numerators begin:
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,
...
Periodic part = [4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1];

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A060819, A061037, A106617, A109008, A188134, A213268.

[Denominator((n+4)/Gcd(n,4)^2): n in [0..100]]; // G. C. Greubel, Aug 05 2018

a[n_] := (n+4)/GCD[n, 4]^2 // Denominator;  Table[a[n], {n, 0, 100}]
(* or: *)
Table[{1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4}[[Mod[n, 16, 1]]], {n, 0, 100}]

for(n=0,100, print1(denominator((n+4)/gcd(n,4)^2), ", ")) \\ G. C. Greubel, Aug 05 2018

A247004 = A060819 / A109008.
(n+4) / gcd(n, 4)^2 = A188134(n+4) / 4. - Michael Somos, Sep 12 2014
a(n) = a(n+16) = a(-n), a(2*n + 1) = 1 for all n in Z. - Michael Somos, Sep 13 2014

A231190 Numerator of abs(n-8)/(2*n), n >= 1.

Original entry on oeis.org

7, 3, 5, 1, 3, 1, 1, 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

Offset: 1

Wolfdieter Lang, Dec 12 2013

Because 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*a(n)/b(n)) with gcd(a(n),b(n)) = 1, one has
  2*sin(Pi*4/n) = R(a(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/b(n)) =: rho(b(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively.
  b(n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(b(n))) of degree delta(b(n)), with delta(k) = A055034(k). This degree delta(b(n)) is given in A231193(n), and if gcd(n,2) = 1 it coincides with the one for sin(2*Pi/n) given by A093819(n). See Theorem 3.9 of the I. Niven reference, pp. 37-38, which uses gcd(k, n) = 1. See also the Jan 09 2011 comment on A093819.
a(n) and b(n) = A232625(n) are the k=2 members of a family of pair of sequences p(k,n) and q(k,n), n >= 1, k >= 1, relevant to determine the algebraic degree of 2*sin(Pi*2*k/n) from the trigonometric identity (used in the D. H. Lehmer and I. Niven references) 2*sin(Pi*2*k/n) = 2*cos(Pi*abs(n-4*k)/(2*n)) = 2*cos(Pi*p(k,n)/q(k,n)). This is R(p(k,n), x) (mod C(q(k,n), x)), with x = 2*cos(Pi/q(k,n)) =: rho(q(k,n)). The polynomials R and C have been used above. C(q(k,n), x) is the minimal polynomial of rho(q(k,n)) with degree delta(q(k,n)), which is then the degree, call it deg(k,n), of the integer 2*sin(Pi*2*k/n) in the number field Q(rho(q(k,n))). From Theorem 3.9 of the I. Niven reference deg(k,n) is, for given k, for those n with gcd(k, n) = 1 determined by A093819(n). In general deg(k,n) = A093819(n/gcd(k,n)). For the k=1 instance p(1,n) and q(1,n) see comments on A106609 and A225975.

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Robert Israel, Table of n, a(n) for n = 1..10000
D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.

Cf. A127672 (R), A187360 (C), A232625 (b), A055034 (delta), A093819 (degree if k=1), A232626(degree if k=2), A106609 (k=1, p), A225975 (k=1, q), A106617.

f:= n -> numer(abs(n-8)/(2*n)):
map(f, [$1..100]); # Robert Israel, Dec 06 2018

a[n_] := Numerator[Abs[n-8]/(2n)]; Array[a, 50] (* Amiram Eldar, Dec 06 2018 *)

a(n) = numerator(abs(n-8)/(2*n)), n >= 1.
a(n) = abs(n-8)/gcd(n-8, 16).
a(n) = abs(n-8) if n is odd; if n is even then a(n) = abs(n-8)/2 if n/2 == 1, 3, 5, 7 (mod 8), a(n) = abs(n-8)/4  if n/2 == 2, 6 (mod 8), a(n) = abs(n-8)/8 if n/2 == 0 (mod 8) and a(n) = abs(n-8)/16 if n == 4 (mod 8).
O.g.f.: 1+ x*(7 + 3*x + 5*x^2 + 1*x^3 + 3*x^4 + 1*x^5 + 1*x^6) + N(x)/(1-x^16)^2 , with N(x) = x^9*((1+x^30) + x*(1+x^28)  + 3*x^2*(1+x^26) + x^3*(1+x^24) + 5*x^4*(1+x^22) + 3*x^5*(1+x^20) + 7*x^6*(1+x^18) + x^7*(1+x^16) + 9*x^8*(1+x^14) + 5*x^9*(1+x^12) + 11*x^10*(1+x^10) + 3*x^11*(1+x^8) + 13*x^12*(1+x^6) + 7*x^13*(1+x^4) + 15*x^14*(1+x^2)+x^15).
a(n+32)-2*a(n+16)+a(n) = 0 for n >= 8.
a(n+8) = A106617(n). - Peter Bala, Feb 28 2019

A213268 Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 2, 2, 8, 2, 1, 1, 4, 1, 16, 16, 1, 2, 1, 8, 8, 32, 16, 1, 1, 4, 4, 16, 8, 64, 64, 1, 2, 2, 8, 4, 32, 32, 128, 16, 1, 1, 4, 2, 16, 16, 64, 8, 256, 256, 1, 2, 1, 8, 8, 32, 4, 128, 128, 512, 256, 1, 1, 4, 4, 16, 2, 64, 64, 256, 128, 1024, 1024

Offset: 0

Paul Curtz, Jun 08 2012

Starting from any sequence a(k) in the first row, define the array T(n,k) of the inverse semi-binomial transform  by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -T(n-1,k)/2, n>=1.
Here, where the first row is the nonnegative integers, the array is
0        1      2      3    4      5      6      7     8   =A001477(n)
1       3/2     2    5/2    3    7/2      4    9/2     5   =A026741(n+2)/A000034(n)
1       5/4   3/2    7/4    2    9/4    5/2   11/4     3   =A060819(n+4)/A176895(n)
3/4     7/8     1    9/8  5/4   11/8    3/2   13/8   7/4   =A106609(n+6)/A205383(n+6)
1/2    9/16   5/8  11/16  3/4  13/16    7/8  15/16     1   =A106617(n+8)/TBD
5/16  11/32   3/8  13/32 7/16  15/32    1/2  17/32  9/16
3/16  13/64  7/32  15/64  1/4  17/64   9/32  19/64  5/16
7/64 15/128   1/8 17/128 9/64 19/128   5/32 21/128 11/64
1/16 17/256 9/128 19/256 5/64 21/256 11/128 23/256  3/32.
The first column contains 0, followed by fractions A000265/A084623, that is Oresme numbers n/2^n multiplied by 2 (see A209308).

			The array of denominators starts:
  1   1   1   1   1   1   1   1   1   1   1 ...
  1   2   1   2   1   2   1   2   1   2   1 ...
  1   4   2   4   1   4   2   4   1   4   2 ...
  4   8   1   8   4   8   2   8   4   8   1 ...
  2  16   8  16   4  16   8  16   1  16   8 ...
16  32   8  32  16  32   2  32  16  32   8 ...
16  64  32  64   4  64  32  64  16  64  32 ...
64 128   8 128  64 128  32 128  64 128  16 ...
16 256 128 256  64 256 128 256  32 256 128 ...
256 512 128 512 256 512  64 512 256 512 128 ...
All entries are powers of 2.

A213268frac := proc(n,k)
        if n = 0 then
                return k ;
        else
                return procname(n-1,k+1)-procname(n-1,k)/2 ;
        end if;
end proc:
A213268 := proc(n,k)
        denom(A213268frac(n,k)) ;
end proc: # R. J. Mathar, Jun 30 2012

T[0, k_] := k; T[n_, k_] := T[n, k] = T[n-1, k+1] - T[n-1, k]/2; Table[T[n-k, k] // Denominator, {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2014 *)

A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

Original entry on oeis.org

2, 3, 1, 4, 5, 2, 5, 3, 7, 1, 6, 7, 8, 9, 2, 7, 2, 1, 5, 11, 1, 8, 9, 10, 11, 12, 13, 2, 9, 5, 11, 3, 13, 7, 15, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1

Offset: 1

Wolfdieter Lang, Feb 21 2013

The triangle of the corresponding denominators is given in A221918.
See A221918 for comments on resistance, reduced mass and radius of the twin circles in Archimedes's arbelos, as well as references.
The column sequences give A000027(n+1), A060819(n+2), A106610(n+3), A106617(n+4), A132739(n+5), A222464 for n >= m = 1,2,..., 6.

			The triangle a(n,m) begins:
n\m   1   2    3   4    5   6    7    8   9   10  11  12 ...
1:    2
2:    3   1
3:    4   5    2
4:    5   3    7   1
5:    6   7    8   9    2
6:    7   2    1   5   11   1
7:    8   9   10  11   12  13    2
8:    9   5   11   3   13   7   15    1
9:   10  11    4  13   14   5   16   17
10:  11   3   13   7    3   4   17    9  19    1
11:  12  13   14  15   16  17   18   15  20   21   2
12:  13   7    5   1   17   1   19    5   7   11  23   1
...
a(n,1) = n + 1 because R(n,1) = n/(n+1), gcd(n,n+1) = 1, hence denominator(R(n,m)) = n + 1.
a(5,4) = 9 because R(5,4) = 20/9, gcd(20,9) = 1, hence denominator( R(5,4)) = 9.
a(6,3) = 1 because R(6,3) = 18/9 = 2/1.
For the rationals R(n,m) see A221918.

Cf. A221918 (companion triangle).

a[n_, m_] := Numerator[1/n + 1/m]; Table[a[n, m], {n, 1, 12}, {m, 1, n}] // Flatten  (* Jean-François Alcover, Feb 25 2013 *)

a(n,m) = numerator(2/n + 1/m), n >= m >= 1, and 0 otherwise.

A221918(n,m)/a(n,m) = R(n,m) = n*m/(n+m). 1/R(n,m) = 1/n + 1/m.

A227140 a(n) = n/gcd(n,2^5), n >= 0.

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, 1, 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, 2, 65, 33, 67, 17, 69, 35

Offset: 0

Wolfdieter Lang, Jul 04 2013

H(n,4) = 2*n*4/(n+4) is the harmonic mean of n and 4. For n >= 4 the denominator of H(n,4) is (n+4)/gcd(8*n,n+4) = (n+4)/gcd(n+4,32). a(n+8) = A227042(n+4,4), n >= 0. The numerator of H(n,4) is given in A227107. Thus a(n) is related to denominator of the harmonic mean H(n-4, 4).
Note the difference from A000265(n) (odd part of n) = n/gcd(n,2^n), n >= 1, which differs for the first time for n = 64. a(64) = 2, not 1.
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 27 2019

Andrew Howroyd, Table of n, a(n) for n = 0..1000
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, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A000265, A227042, A227107, A106617, A276234.

List([0..80], n-> n/Gcd(n, 2^5)); # G. C. Greubel, Feb 27 2019

[n/GCD(n, 2^5): n in [0..80]]; // G. C. Greubel, Feb 27 2019

seq(n/igcd(n,32),n=0..70); # Muniru A Asiru, Feb 28 2019

With[{c=2^5},Table[n/GCD[n,c],{n,0,70}]] (* Harvey P. Dale, Feb 16 2018 *)

a(n)=n/gcd(n, 2^5); \\ Andrew Howroyd, Jul 23 2018

[n/gcd(n,2^5) for n in (0..80)] # G. C. Greubel, Feb 27 2019

a(n) = n/gcd(n, 2^5).
a(n) = denominator(8*(n-4)/n), n >= 0 (with denominator(infinity) = 0).
From Peter Bala, Feb 27 2019: (Start)
a(n) = numerator(n/(n + 32)).
O.g.f.: F(x) - F(x^2) - F(x^4) - F(x^8) - F(x^16) - F(x^32), where F(x) = x/(1 - x)^2. Cf. A106617. (End)
From Bernard Schott, Mar 02 2019: (Start)
a(n) = 1 iff n is 1, 2, 4, 8, 16, 32 and a(2^n) = 2^(n-5) for n >= 5.
a(n) = n iff n is odd (A005408). (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^(e-min(e,5)), and a(p^e) = p^e for p > 2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/2^(2*s) - 1/2^(3*s) - 1/2^(4*s) - 1/2^(5*s)).
Sum_{k=1..n} a(k) ~ (683/2048) * n^2. (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

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.

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..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)

A225055 Irregular triangle which lists the three positions of 2*n-1 in A060819 in row n.

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 5, 10, 20, 7, 14, 28, 9, 18, 36, 11, 22, 44, 13, 26, 52, 15, 30, 60, 17, 34, 68, 19, 38, 76, 21, 42, 84, 23, 46, 92, 25, 50, 100, 27, 54, 108, 29, 58, 116, 31, 62, 124, 33, 66, 132, 35, 70, 140, 37, 74, 148

Offset: 1

Paul Curtz, Apr 26 2013

There are no multiples of 8 in the triangle.
A047592 contains a sorted list of all elements of the triangle.
The triangle is a member of a family of triangles with parameter k that list the k positions of 2*n-1: 2*n-1 in A000027 (k=1), A043547 the k=2 positions in A026741, the triangle 1,2,4,8; 3,6,12,24;... with the k=4 positions in A106609, or the triangle 1,2,4,8,16; 3,6,12,24,48;... with the k=5 positions in A106617.

			1, 2, 4;  # 1 at A060819(1), A060819(2) and A060819(4)
3, 6, 12;  # 3 at A060819(3), A060819(6) and A060819(12)
5, 10, 20;
7, 14, 28;
9, 18, 36;
11, 22, 44;
13, 26, 52;
15, 30, 60;

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A147587. A042968.

numberOfTriplets = 19; A060819 = Table[n/GCD[n, 4], {n, 1, 8*numberOfTriplets}]; Table[Position[A060819, 2*n-1], {n, 1, numberOfTriplets}] // Flatten (* Jean-François Alcover, Apr 30 2013 *)

T(n,1) = 2*n-1. T(n,2) = 4*n-2. T(n,3) = 8*n-4.

A276234 a(n) = n/gcd(n, 256).

Original entry on oeis.org

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, 1, 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, 1, 65, 33, 67, 17, 69, 35, 71

Offset: 1

Artur Jasinski, Aug 24 2016

a(n) first differs from A000265(n) at n = 512. - Andrew Howroyd, Jul 23 2018

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 27 2019

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, order 512.

Cf. A276233 (numerators), A227140, A000265, A106617.

List([1..80], n-> n/Gcd(n, 2^8)); # G. C. Greubel, Feb 27 2019

[n/GCD(n, 2^8): n in [1..80]]; // G. C. Greubel, Feb 27 2019

seq(n/igcd(n,256), n=1..100); # Robert Israel, Aug 26 2016

Table[n/GCD[n, 2^8], {n,1,80}] (* G. C. Greubel, Feb 27 2019 *)

a(n)=n/gcd(n,256) \\ Charles R Greathouse IV, Aug 26 2016

[n/gcd(n, 2^8) for n in (1..80)] # G. C. Greubel, Feb 27 2019

a(2k-1) = 2k-1.
G.f.: (x+x^3)/(1-x^2)^2 +(x^2+x^6)/(1-x^4)^2 +(x^4+x^12)/(1-x^8)^2 +(x^8+x^24)/(1-x^16)^2 +(x^16+x^48)/(1-x^32)^2 +(x^32+x^96)/(1-x^64)^2 +(x^64+x^192)/(1-x^128)^2 +(x^128+x^256+x^384)/(1-x^256)^2. - Robert Israel, Aug 26 2016
a(n) = 2*a(n-256) - a(n-512). - Charles R Greathouse IV, Aug 26 2016
From Peter Bala, Feb 27 2019: (Start)
a(n) = numerator(n/(n + 256)).
O.g.f.: F(x) - Sum_{k = 1..8} F(x^(2^k)), where F(x) = x/(1 - x)^2. Cf. A106617. (End)
From Amiram Eldar, Nov 26 2022: (Start)
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/2^(2*s) - 1/2^(3*s) - 1/2^(4*s) - 1/2^(5*s) - 1/2^(6*s) - 1/2^(7*s) - 1/2^(8*s)).
Multiplicative with a(2^e) = 2^(e-min(e,8)), and a(p^e) = p^e for p > 2.
Sum_{k=1..n} a(k) ~ (43691/131072) * n^2. (End)

Keyword:mult added and terms a(51) and beyond from Andrew Howroyd, Jul 23 2018

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).

cp's OEIS Frontend

A247004 Denominator of (n+4)/gcd(n, 4)^2, a 16-periodic sequence that associates A061037 with A106617.

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A231190 Numerator of abs(n-8)/(2*n), n >= 1.

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A213268 Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.

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A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

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A227140 a(n) = n/gcd(n,2^5), n >= 0.

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A106616 Numerator of n/(n+15).

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A225055 Irregular triangle which lists the three positions of 2*n-1 in A060819 in row n.

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