A106609 -id:A106609 - cp's OEIS frontend

A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 1, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 55, 7, 57, 29, 59, 15, 61, 31, 21, 1, 65, 33, 67, 17, 69, 35, 71, 3

Offset: 1

Wolfdieter Lang, Jan 19 2012

There are no odd primes dividing n iff n is a power of 2.
This sequence coincides with the bisection of A007947 (even indices), which is A099985, dividing out the even prime 2 in the squarefree kernel.
a(n) divides A106609(n) for n>=1. - Alexander R. Povolotsky, Apr 06 2015

			a(5)=5 because 5 is a single odd prime.
a(9)=3 because 9=3*3 has as squarefree part 3.
a(1)=1 because 1 is a power of 2, having no odd primes as a factor.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A099985, A099984, A007947, A000265, A065463, A106609.

A204455 := proc(n)
    local p;
    numtheory[factorset](n) minus {2} ;
    mul(p,p=%) ;
end proc:
seq(A204455(n),n=1..40) ; # R. J. Mathar, Jan 25 2017

f[n_] := Select[First /@ FactorInteger@ n, PrimeQ@ # && OddQ@ # &]; Times @@@ (f /@ Range@ 120) (* Michael De Vlieger, Apr 08 2015 *)

a(n) = {my(f = factor(n)); prod(k=1, #f~, if (f[k,1] % 2, f[k,1], 1));} \\ Michel Marcus, Apr 07 2015

a(n) = factorback(setminus(factorint(n)[, 1]~, [2])) \\ Jianing Song, Aug 09 2022

a(n) = A099985(n)/2 = A007947(2*n)/2.
a(n) = A000265(A007947(n)) = A007947(A000265(n)). - Charles R Greathouse IV, Jan 19 2012
Multiplicative with a(p^e)=p for p <> 2 and a(2^e)=1. - R. J. Mathar, Jul 02 2013
a(n) = Sum_{d|n} phi(d)*mu(2d)^2. - Ridouane Oudra, Sep 02 2019
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} mu(2*n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(2*gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (2/5) * A065463 = 0.281776... . - Amiram Eldar, Nov 19 2022
a(n) = Sum_{d divides n, d odd} mu(d)^2 * phi(d). - Peter Bala, Feb 01 2024

A106608 a(n) = numerator of n/(n+7).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 3, 22, 23, 24, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 5, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 10, 71, 72, 73, 74, 75, 76

Offset: 0

N. J. A. Sloane, May 15 2005

In general, the numerators of n/(n+p) for prime p and n >= 0 form a sequence with the g.f.: x/(1-x)^2 - (p-1)*x^p/(1-x^p)^2. - Paul D. Hanna, Jul 27 2005
Generalizing the above comment of Hanna, the numerators of n/(n + k) for a fixed positive integer k and n >= 0 form a sequence with the g.f.: Sum_{d divides k} f(d)*x^d/(1 - x^d)^2, where f(n) is the Dirichlet inverse of the Euler totient function A000010. f(n) is a multiplicative function defined on prime powers p^k by f(p^k) = 1 - p. See A023900. - Peter Bala, Feb 17 2019
a(n) <> n iff n = 7 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019

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

Cf. A000010, A109048, A023900.

Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106609 thru A106612 (k = 8 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([0..80],n->NumeratorRat(n/(n+7))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+7)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+7)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+7)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)

vector(100, n, n--; numerator(n/(n+7))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,7)/7 for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009

G.f.: x/(1-x)^2 - 6*x^7/(1-x^7)^2. - Paul D. Hanna, Jul 27 2005
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109048(n)/7.
Dirichlet g.f.: zeta(s-1)*(1-6/7^s).
Multiplicative with a(p^e) = p^e if p<>7, a(7^e) = 7^(e-1) if e > 0. (End)
Sum_{k=1..n} a(k) ~ (43/98) * n^2. - Amiram Eldar, Nov 25 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 13*log(2)/7. - Amiram Eldar, Sep 08 2023

A125650 Numerator of n(n+3)/(4(n+1)(n+2)) = sum(k=1..n, 1/(k(k+1)(k+2)) ).

Original entry on oeis.org

0, 1, 5, 9, 7, 5, 27, 35, 11, 27, 65, 77, 45, 26, 119, 135, 38, 85, 189, 209, 115, 63, 275, 299, 81, 175, 377, 405, 217, 116, 495, 527, 140, 297, 629, 665, 351, 185, 779, 819, 215, 451, 945, 989, 517, 270, 1127, 1175, 306, 637, 1325, 1377, 715, 371, 1539, 1595, 413, 855

Offset: 0

Alexander Adamchuk, Nov 29 2006

3^2 divides a(3k). p divides a(p) for an odd prime p. p divides a(p-3) for prime p>3. p^k divides a(p^k) for an odd prime p. a(n) = m^2 is a perfect square for n = {1,3,24,147,864,5043,29400,171363,...} = A125651(n). Corresponding numbers m such that m^2 = a[ A125651(n) ] are listed in A125652(n) = {1,3,9,105,306,3567,10395,121173,...}.

			The rationals n(n+3)/(4(n+1)(n+2)) = a(n)/A230328(n) begin:
0, 1/6, 5/24, 9/40, 7/30, 5/21, 27/112, 35/144, 11/45, 27/110, 65/264, 77/312, 45/182, 26/105, 119/480, ... - _Wolfdieter Lang_, Mar 08 2018

L. B.W. Jolley, Summation of Series, Second revised ed., Dover, 1961, p.38, (201).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A125651, A125652. A160050, A230328 (denominators).

[Numerator(n*(n+3)/(4*(n+1)*(n+2))): n in [0..60]]; // Vincenzo Librandi, May 21 2012

Table[Numerator[n(n+3)/(4(n+1)(n+2))],{n,0,100}]

a(n)=n*(n+3)/2^min(3,valuation(n*(n+3),2)); \\ Max Alekseyev, Jan 11 2007

a(n) = Sum_{k=1..n} 1/(k(k+1)(k+2)).
a(n) = n*(n+3)/2^min(3,valuation(n*(n+3),2)). a(n)=n*(n+3)/4 for n=1 or 4 (mod 8); a(n)=n*(n+3)/8 for n=0 or 5 (mod 8); a(n) = n*(n+3)/2 for n=2, 3, 6, or 7 (mod 8). - Max Alekseyev, Jan 11 2007
a(n) = A106609(n)*A106609(n+3). - Paul Curtz, Jan 13 2011
G.f.: x*(x^19 -2*x^18 +3*x^17 -5*x^16 +3*x^15 -6*x^14 +7*x^13 -11*x^12 -12*x^11 +24*x^10 -36*x^9 +24*x^8 -38*x^7 +28*x^6 -18*x^5 -3*x^4 -2*x -1) / ((x-1)^3*(x^2+1)^3*(x^4+1)^3). - Colin Barker, Feb 21 2013
G.f. for rationals r(n) = a(n)/A230328(n): (1/4)*(1 - hypergeometric([1, 2], [3], -x/(1-x)))/(1-x) = (- 2*x + 3*x^2 + 2*(2*x - (1 + x^2))*log(1-x))/(4*(1-x)*x^2). For the r(n) formula see Jolley's general remark (201) on p.38. Thanks to Gary Detlefs for pointing to this remark. - Wolfdieter Lang, Mar 08 2018

A225948 a(0) = -1; for n>0, a(n) = numerator(1/4 - 4/n^2).

Original entry on oeis.org

-1, -15, -3, -7, 0, 9, 5, 33, 3, 65, 21, 105, 2, 153, 45, 209, 15, 273, 77, 345, 6, 425, 117, 513, 35, 609, 165, 713, 12, 825, 221, 945, 63, 1073, 285, 1209, 20, 1353, 357, 1505, 99, 1665, 437, 1833, 30, 2009, 525, 2193, 143

Offset: 0

Paul Curtz, May 21 2013

Denominators are in A226008.
Fractions in lowest terms for n>0: -15/4, -3/4, -7/36, 0/1, 9/100, 5/36, 33/196, 3/16, 65/324, 21/100, 105/484, 2/9, 153/676, 45/196, 209/900, 15/64,...
If t(n) is the sequence with period 8: 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, ... (see A226044), then A226008(n) = 4*a(n) + t(n).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,1).

Cf. A000466, A028566, A061037, A061041, A078371, A106609, A142705, A145923, A226008.

[-1] cat [Numerator(1/4-4/n^2): n in [1..50]]; // Bruno Berselli, May 22 2013

Join[{-1}, Table[Numerator[1/4 - 4/n^2], {n, 50}]] (* Bruno Berselli, May 24 2013 *)

concat([-1], vector(100, n, numerator(1/4 - 4/n^2))) \\ G. C. Greubel, Sep 19 2018

a(n) = 3*a(n-8) -3*a(n-16) +a(n-24).
a(2n) = A061037(n), a(2n+1) = A145923(n-2) for A145923(-2)=-15, A145923(-1)=-7.
a(4n)   = A142705(n) for A142705(0)=-1, a(8n) = A000466(n);
a(4n+1) = A028566(4n-3) for A028566(-3)=-15;
a(4n+2) = A078371(n-1) for A078371(-1)=-3;
a(4n+3) = A028566(4n-1) for A028566(-1)=-7.
a(n+4)  = A106609(n) * A106609(n+8).
G.f.: -(1 +15*x +3*x^2 +7*x^3 -9*x^5 -5*x^6 -33*x^7 -6*x^8 -110*x^9 -30*x^10 -126*x^11 -2*x^12 -126*x^13 -30*x^14 -110*x^15 -3*x^16 -33*x^17 -5*x^18 -9*x^19 +7*x^21 +3*x^22 +15*x^23)/(1-x^8)^3. - Bruno Berselli, May 22 2013
a(n) = (n^2-16)*(6*cos(Pi*n/4)-54*cos(Pi*n/2)+6*cos(3*Pi*n/4)-219*(-1)^n+293)/512. - Bruno Berselli, May 22 2013
a(n+10) = a(n+2)*(n+14)/(n-2) for n=0,1 and n>2. - Bruno Berselli, May 22 2013

Edited by Bruno Berselli, May 22 2013

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

A109049 a(n) = lcm(n, 8).

Original entry on oeis.org

0, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 88, 24, 104, 56, 120, 16, 136, 72, 152, 40, 168, 88, 184, 24, 200, 104, 216, 56, 232, 120, 248, 32, 264, 136, 280, 72, 296, 152, 312, 40, 328, 168, 344, 88, 360, 184, 376, 48, 392, 200, 408, 104, 424, 216, 440, 56, 456, 232

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A106609, A109011, A109042.

[Lcm(n,8): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

LCM[Range[0,60],8] (* Harvey P. Dale, May 03 2019 *)

[lcm(n,8)for n in range(0, 59)] # Zerinvary Lajos, Jun 09 2009

a(n) = n*8/gcd(n, 8).
From R. J. Mathar, Apr 18 2011: (Start)
G.f.: 8*x*(x^2-x+1)*(x^12 + 2*x^11 + 4*x^10 + 3*x^9 + 4*x^8 + 4*x^7 + 7*x^6 + 4*x^5 + 4*x^4 + 3*x^3 + 4*x^2 + 2*x + 1)  / ( (x-1)^2 *(1+x)^2 *(x^2+1)^2 *(x^4+1)^2 ).
a(n) = 8*A106609(n) = 8*n/A109011(n). (End)
Sum_{k=1..n} a(k) ~ (43/16) * n^2. - Amiram Eldar, Nov 26 2022

A225975 Square root of A226008(n).

Original entry on oeis.org

0, 2, 2, 6, 1, 10, 6, 14, 4, 18, 10, 22, 3, 26, 14, 30, 8, 34, 18, 38, 5, 42, 22, 46, 12, 50, 26, 54, 7, 58, 30, 62, 16, 66, 34, 70, 9, 74, 38, 78, 20, 82, 42, 86, 11, 90, 46, 94, 24, 98, 50, 102, 13, 106, 54, 110, 28, 114, 58

Offset: 0

Paul Curtz, May 22 2013

Repeated terms of A016825 are in the positions 1,2,3,6,5,10,... (A043547).
From Wolfdieter Lang, Dec 04 2013: (Start)
This sequence a(n), n>=1, appears in the formula 2*sin(2*Pi/n) = R(p(n), x) modulo C(a(n), x), with x = rho(a(n)) = 2*cos(Pi/a(n)), the R-polynomials given in A127672 and the minimal C-polynomials of rho given in A187360. This follows from the identity 2*sin(2*Pi/n) = 2*cos(Pi*p(n)/a(n)) with gcd(p(n), a(n)) = 1. For p(n) see a comment on A106609,
Because R is an integer polynomial it shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(a(n))) of degree delta(a(n)) (the degree of C(a(n), x)), with delta(k) = A055034(k). This degree is given in A093819. For the coefficients of 2*sin(2*Pi/n) in the power basis of Q(rho(a(n))) see A231189 . (End)

			For the first formula: a(0)=-1+1=0, a(1)=-3+5=2, a(2)=-1+3=2, a(3)=-1+7=6, a(4)=0+1=1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

Cf. A016825, A017089, A017137, A022998, A106609, A145979, A226008.

a[0]=0; a[n_] := Sqrt[Denominator[1/4 - 4/n^2]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, May 30 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,2,2,6,1,10,6,14,4,18,10,22,3,26,14,30},60] (* Harvey P. Dale, Nov 21 2019 *)

a(n) = A106609(n-4) + A106609(n+4) with A106609(-4)=-1, A106609(-3)=-3, A106609(-2)=-1, A106609(-1)=-1.
a(n) = 2*a(n-8) -a(n-16).
a(2n+1) = A016825(n), a(2n) = A145979(n-2) for n>1, a(0)=0, a(2)=2.
a(4n)   = A022998(n).
a(4n+1) = A017089(n).
a(4n+2) = A016825(n).
a(4n+3) = A017137(n).
G.f.: x*(2 +2*x +6*x^2 +x^3 +10*x^4 +6*x^5 +14*x^6 +4*x^7 +14*x^8 +6*x^9 +10*x^10 +x^11 +6*x^12 +2*x^13 +2*x^14)/((1-x)^2*(1+x)^2*(1+x^2)^2*(1+x^4)^2). [Bruno Berselli, May 23 2013]
From Wolfdieter Lang, Dec 04 2013: (Start)
a(n) = 2*n if n is odd; if n is even then a(n) is n if n/2 == 1, 3, 5, 7 (mod 8), it is n/2 if n/2 == 0, 4 (mod 8) and it is n/4 if n/2 == 2, 6 (mod 8). This leads to the given G.f..
With c(n) = A178182(n), n>=1, a(n) = c(n)/2 if c(n) is even and c(n) if c(n) is odd. This leads to the preceding formula. (End)

Edited by Bruno Berselli, May 24 2013

A232625 Denominators of abs(n-8)/(2*n), n >= 1.

Original entry on oeis.org

2, 2, 6, 2, 10, 6, 14, 1, 18, 10, 22, 6, 26, 14, 30, 4, 34, 18, 38, 10, 42, 22, 46, 3, 50, 26, 54, 14, 58, 30, 62, 8, 66, 34, 70, 18, 74, 38, 78, 5, 82, 42, 86, 22, 90, 46, 94, 12, 98, 50, 102, 26, 106, 54, 110, 7, 114, 58, 118, 30, 122, 62, 126, 16, 130, 66

Offset: 1

Wolfdieter Lang, Dec 12 2013

The numerators are given in A231190. See the comments there on 2*sin(Pi*4/n).
2*sin(Pi*4/n) = R(b(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/a(n)) =: rho(a(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively.  b(n) = A231190(n).
delta(a(n)) = deg(2,n), with delta(k) = A055034(k), is the degree of the algebraic number 2*sin(Pi*4/n) given in A232626.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A127672 (R), A187360 (C), A231190 (b), A055034 (delta), A232626 (degree k=2), A106609 (k=1, p), A225975 (k=1, q), A093819 (degree k=1).

a[n_] := Denominator[(n-8)/(2*n)]; Array[a, 100] (* Amiram Eldar, Nov 09 2024 *)

a(n) = denominator((n-8)/(2*n)); \\ Amiram Eldar, Nov 09 2024

a(n) = denominator(abs(n-8)/(2*n)), n >= 1.
a(n) = 2*n/gcd(n-8, 16).
a(n) = 2*n if n is odd; if n is even then a(n) = n if n/2 == 1, 3, 5, 7 (mod 8), a(n) = n/2  if n/2 == 2, 6 (mod 8), a(n) == n/4 if n/2 == 0 (mod 8) and a(n) = n/8 if n == 4 (mod 8).
O.g.f.: x*(2*(1+x^30) + 2*x*(1+x^28) + 6*x^2*(1+x^26) + 2*x^3*(1+x^24) + 10*x^4*(1+x^22) + 6*x^5*(1+x^20) + 14*x^6*(1+x^18) + x^7*(1+x^16) + 18*x^8*(1+x^14) + 10*x^9*(1+x^12) + 22*x^10*(1+x^10) + 6*x^11*(1+x^8) + 26*x^12*(1+x^6) + 14*x^13*(1+x^4) + 30*x^14*(1+x^2) + 4*x^15)/(1-x^16)^2.
Sum_{k=1..n} a(k) ~ (171/256) * n^2. - Amiram Eldar, Nov 09 2024

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 *)

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.

cp's OEIS Frontend

A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A106608 a(n) = numerator of n/(n+7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A125650 Numerator of n(n+3)/(4(n+1)(n+2)) = sum(k=1..n, 1/(k(k+1)(k+2)) ).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A225948 a(0) = -1; for n>0, a(n) = numerator(1/4 - 4/n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A109049 a(n) = lcm(n, 8).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula