A112329 -id:A112329 - cp's OEIS frontend

A246063 First occurrence of n in sequence A112329.

2, 1, 3, 9, 15, 64, 45, 256, 96, 144, 192, 4096, 240, 16384, 768, 576, 480, 262144, 720, 1048576, 960, 2304, 12288, 16777216, 1440, 5184, 49152, 3600, 3840, 1073741824, 2880, 4294967296, 3360, 36864, 786432, 20736, 5040, 274877906944, 3145728, 147456, 6720

Offset: 0

Ray Chandler, Aug 24 2014

nonn

Inspired by a comment from Robert G. Wilson v in sequence A112329.

Ray Chandler, Table of n, a(n) for n = 0..3322 [terms <= 1000 digits]

Cf. A094572, A112329.

g[lst_,p_]:=Module[{t,i,j},Union[Flatten[Table[t=lst[[i]];t[[j]]=p*t[[j]];Sort[t],{i,Length[lst]},{j,Length[lst[[i]]]}],1],Table[Sort[Append[lst[[i]],p]],{i,Length[lst]}]]];f[n_]:=Module[{i,j,p,e,lst={{}}},{p,e}=Transpose[FactorInteger[n]];Do[lst=g[lst,p[[i]]],{i,Length[p]},{j,e[[i]]}];lst];
(* above factor functions from T. D. Noe in A162247 *)
nmax=100;
a1={2,1,3};
Do[
least=Infinity;
fn=f[n];
Do[
exps=Reverse[fnitem]-1;
odd=even=1;
cnt=0;
Do[
cnt++;
odd*=(Prime[cnt+1]^exp);
even*=(Prime[cnt]^exp);
,{exp,exps}];
least=Min[least,odd,4even];
,{fnitem,fn}];
AppendTo[a1,least];
,{n,3,nmax}];
a1

d(n) = if (denominator(n)==1, numdiv(n), 0);
f(n) = numdiv(n) - 2*d(n/2) + 2*d(n/4);
a(n) = {my(k = 1); while (f(k) != n, k++); k;} \\ Michel Marcus, Jul 30 2017

a(p) = 2^(p+1) for prime p >= 5.

A048272 Number of odd divisors of n minus number of even divisors of n.

Original entry on oeis.org

1, 0, 2, -1, 2, 0, 2, -2, 3, 0, 2, -2, 2, 0, 4, -3, 2, 0, 2, -2, 4, 0, 2, -4, 3, 0, 4, -2, 2, 0, 2, -4, 4, 0, 4, -3, 2, 0, 4, -4, 2, 0, 2, -2, 6, 0, 2, -6, 3, 0, 4, -2, 2, 0, 4, -4, 4, 0, 2, -4, 2, 0, 6, -5, 4, 0, 2, -2, 4, 0, 2, -6, 2, 0, 6, -2, 4, 0, 2, -6, 5, 0, 2, -4, 4, 0, 4, -4, 2, 0, 4, -2, 4

Offset: 1

Adam Kertesz

abs(a(n)) = (1/2) * (number of pairs (i,j) satisfying n = i^2 - j^2 and -n <= i,j <= n). - Benoit Cloitre, Jun 14 2003
As A001227(n) is the number of ways to write n as the difference of 3-gonal numbers, a(n) describes the number of ways to write n as the difference of e-gonal numbers for e in {0,1,4,8}. If pe(n):=(1/2)*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 4*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=1, 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e in {0,4} and for a(n) itself is a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=8. (Same for e=-1 see A035218.) - Volker Schmitt (clamsi(AT)gmx.net), Nov 09 2004
An argument by Gareth McCaughan suggests that the average of this sequence is log(2). - Hans Havermann, Feb 10 2013 [Supported by a graph. - Vaclav Kotesovec, Mar 01 2023]
From Keith F. Lynch, Jan 20 2024: (Start)
a(n) takes every possible integer value, positive, negative, and zero. Proof: For all nonnegative integers k, a(3^k) = 1+k, a(2^k) = 1-k.
a(n) takes every possible integer value except 1 and -1 infinitely many times. Proof: a(o^(k-1)) = k and a(4*o^(k-1)) = -k for all positive integers k and odd primes o, of which there are infinitely many.  a(n) = 0 iff n = 2 (mod 4).  a(n) = 1 iff n = 1.  a(n) = -1 iff n = 4.
a(n) takes prime value p only for n = o^(p-1), where o is any odd prime.
Terms have a simple pattern that repeats with a period of 4: Positive, zero, positive, negative.
(End)
Inverse Möbius transform of (-1)^(n+1). - Wesley Ivan Hurt, Jun 22 2024

			a(20) = -2 because 20 = 2^2*5^1 and (1-2)*(1+1) = -2.
G.f. = x + 2*x^3 - x^4 + 2*x^5 + 2*x^7 - 2*x^8 + 3*x^9 + 2*x^11 - 2*x^12 + ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), first formula.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 97, 7(ii).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n, for n = 1..1000000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_{o-e}(n).
Index entries for sequences mentioned by Glaisher.

Cf. A048298. A diagonal of A060184.
First differences of A059851.
Cf. A001227, A035218, A094572, A099774, A112329, A183063, A000005.
Indices of records: A053624 (highs), A369151 (lows).

a048272 n = a001227 n - a183063 n  -- Reinhard Zumkeller, Jan 21 2012

[&+[(-1)^(d+1):d in Divisors(n)] :n in [1..95] ]; // Marius A. Burtea, Aug 10 2019

add(x^n/(1+x^n), n=1..60): series(%,x,59);
A048272 := proc(n)
    local a;
    a := 1 ;
    for pfac in ifactors(n)[2] do
        if pfac[1] = 2 then
            a := a*(1-pfac[2]) ;
        else
            a := a*(pfac[2]+1) ;
        end if;
    end do:
    a ;
end proc: # Schmitt, sign corrected R. J. Mathar, Jun 18 2016
# alternative Maple program:
a:= n-> -add((-1)^d, d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2018

Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
dif[n_]:=Module[{divs=Divisors[n]},Count[divs,?OddQ]-Count[ divs, ?EvenQ]]; Array[dif,100] (* Harvey P. Dale, Aug 21 2011 *)
a[n]:=Sum[-(-1)^d,{d,Divisors[n]}] (* Steven Foster Clark, May 04 2018 *)
f[p_, e_] := If[p == 2, 1 - e, 1 + e]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 09 2022 *)

{a(n) = if( n<1, 0, -sumdiv(n, d, (-1)^d))}; /* Michael Somos, Jul 22 2006 */

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j); \\ log case
s=-log(prod(j=1,N,(1+x^j)^(1/j)));
s=serconvol(s,c)
v=Vec(s) \\ Joerg Arndt, May 03 2008

a(n)=my(o=valuation(n,2),f=factor(n>>o)[,2]);(1-o)*prod(i=1,#f,f[i]+1) \\ Charles R Greathouse IV, Feb 10 2013

a(n)=direuler(p=1,n,if(p==2,(1-2*X)/(1-X)^2,1/(1-X)^2))[n] /* Ralf Stephan, Mar 27 2015 */

{a(n) = my(d = n -> if(frac(n), 0, numdiv(n))); if( n<1, 0, if( n%4, 1, -1) * (d(n) - 2*d(n/2) + 2*d(n/4)))}; /* Michael Somos, Aug 11 2017 */

Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n) = Sum_{n >= 1} (-1)^(n-1)*x^n/(1-x^n). Expand Sum 1/(1+x^n) in powers of 1/x.
If n = 2^p1*3^p2*5^p3*7^p4*11^p5*..., a(n) = (1-p1)*Product_{i>=2} (1+p_i).
Multiplicative with a(2^e) = 1 - e and a(p^e) = 1 + e if p > 2. - Vladeta Jovovic, Jan 27 2002
a(n) = (-1)*Sum_{d|n} (-1)^d. - Benoit Cloitre, May 12 2003
Moebius transform is period 2 sequence [1, -1, ...]. - Michael Somos, Jul 22 2006
G.f.: Sum_{k>0} -(-1)^k * x^(k^2) * (1 + x^(2*k)) / (1 - x^(2*k)) [Ramanujan]. - Michael Somos, Jul 22 2006
Equals A051731 * [1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 07 2007
From Reinhard Zumkeller, Jan 21 2012: (Start)
a(n) = A001227(n) - A183063(n).
a(A008586(n)) < 0; a(A005843(n)) <= 0; a(A016825(n)) = 0; a(A042968(n)) >= 0; a(A005408(n)) > 0. (End)
a(n) = Sum_{k=0..n} A081362(k)*A015723(n-k). - Mircea Merca, Feb 26 2014
abs(a(n)) = A112329(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014
From Peter Bala, Jan 07 2015: (Start)
Logarithmic g.f.: log( Product_{n >= 1} (1 + x^n)^(1/n) ) = Sum_{n >= 1} a(n)*x^n/n.
a(n) = A001227(n) - A183063(n). By considering the logarithmic generating functions of these three sequences we obtain the identity
( Product_{n >= 0} (1 - x^(2*n+1))^(1/(2*n+1)) )^2 = Product_{n >= 1} ( (1 - x^n)/(1 + x^n) )^(1/n). (End)
Dirichlet g.f.: zeta(s)*eta(s) = zeta(s)^2*(1-2^(-s+1)). - Ralf Stephan, Mar 27 2015
a(2*n - 1) = A099774(n). - Michael Somos, Aug 12 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) - x^k)^(n-k)  =  Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) + x^k)^(n-k) * (-1)^k  =  Sum_{n>=0} a(n)*x^(2*n). (End)
a(n) = 2*A000005(2n) - 3*A000005(n). - Ridouane Oudra, Oct 15 2019
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 2*log(2)-1. - Amiram Eldar, Mar 01 2023

New definition from Vladeta Jovovic, Jan 27 2002

A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, 1, 1, 7, 10, 5, 2, 1, 15, 28, 19, 6, 0, 1, 31, 82, 71, 26, 4, 2, 1, 63, 244, 271, 126, 30, 8, 2, 1, 127, 730, 1055, 626, 196, 50, 13, 3, 1, 255, 2188, 4159, 3126, 1230, 344, 83, 13, 0, 1, 511, 6562, 16511, 15626, 7564, 2402, 583, 91, 6, 2, 1, 1023, 19684, 65791, 78126, 45990, 16808, 4367, 757, 78, 12, 2

Offset: 1

Wolfdieter Lang, Jan 10 2017

The array A(k, n) = sigma^*A279395)%20=%20Sum">(k)(n) (notation of the Hardy reference, given also in a comment in A279395) = Sum{ d >= 1, d divides n} (-1)^(n-d)*d^k, for k >= 0 and n >=1, has the rows A112329, A113184, A064027, A008457, A279395, for k=0..4.
The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.
The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.
The row sums are given in A279397.
The column sums (with offset 0) are A000012, A000225, A034472, A099393, A034474, .. with o.g.f. G(m, z) = (-1)^m*Sum_{d  | m} (-1)^d/(1 - d*z), m >= 1.

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   0
3:  1   1    2
4:  1   3    4    1
5:  1   7   10    5    2
6:  1  15   28   19    6    0
7:  1  31   82   71   26    4   2
8:  1  63  244  271  126   30   8  2
9:  1 127  730 1055  626  196  50 13  3
10: 1 255 2188 4159 3126 1230 344 83 13  0
...
n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,
n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.
n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,
n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,
n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.
...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.

T(n, m) = Sum_{ d >= 1, d divides m} (-1)^(m-d)*d^(n-m) = sigma^*_(n-m)(m), n >= 1, m = 1,2, ..., n. For the definition of
sigma^*_(k)(n) see the Hardy reference or a comment in A279395.
O.g.f triangle T: G(z, x) = Sum_{m>=0}
G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.

A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

Original entry on oeis.org

1, 15, 82, 271, 626, 1230, 2402, 4367, 6643, 9390, 14642, 22222, 28562, 36030, 51332, 69903, 83522, 99645, 130322, 169646, 196964, 219630, 279842, 358094, 391251, 428430, 538084, 650942, 707282, 769980, 923522, 1118479, 1200644, 1252830, 1503652, 1800253, 1874162, 1954830, 2342084, 2733742

Offset: 1

Wolfdieter Lang, Jan 09 2017

This is the k=4 member of the family sigma^*_k(n), defined in the Hardy reference, which is sigma_k(2*j+1) if n = 2*j+1 and sigma_k^e(2*j) - sigma_k^o(2*j) if n=2*j, where the superscript e and o stands for a restriction to even and odd divisors in the sum of their k-th powers, respectively.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher

Cf. A112329 (k=0), A113184 (k=1), A064027 (k=2), A008457(k=3).

[&+[(-1)^(n-d)*d^4:d in Divisors(n)]:n in [1..40]]; // Marius A. Burtea, Aug 17 2019

# A version with signs - N. J. A. Sloane, Nov 23 2018
zet1:=(n,i)->add((-1)^(d-1)*d^i, d in divisors(n));
szet1:=i->[seq(zet1(n,i),n=1..120)];
szet1(4);

f[p_, e_] := If[p == 2, (2^(4*(e + 1)) - 31)/15, (p^(4*(e + 1)) - 1)/(p^4 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 40] (* Amiram Eldar, Aug 17 2019 *)

a(n) = sumdiv(n, d, (-1)^(n-d)*d^4); \\ Michel Marcus, Jan 09 2017

a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.
Bisection: a(2*j-1) = A001159(2*j-1), a(2*j) = 16*A001159(j) - A051001(j), j >= 1. See the comment above for k=4, and the Hardy reference.
G.f.: Sum_{k>=1} k^4*x^k/(1-(-x)^k).
Multiplicative with a(2^k) = 2^4*(2^(4*k)-1)/(2^4-1) - 1 = (2^(4*(k+1)) - 31)/15 and a(p^k) = (p^(4*(k+1))-1)/(p^4-1) for primes p > 2 (see A001159).

A143520 a(n) is n times number of divisors of n if n is odd, zero if n is twice odd, n times number of divisors of n/4 if n is divisible by 4.

Original entry on oeis.org

1, 0, 6, 4, 10, 0, 14, 16, 27, 0, 22, 24, 26, 0, 60, 48, 34, 0, 38, 40, 84, 0, 46, 96, 75, 0, 108, 56, 58, 0, 62, 128, 132, 0, 140, 108, 74, 0, 156, 160, 82, 0, 86, 88, 270, 0, 94, 288, 147, 0, 204, 104, 106, 0, 220, 224, 228, 0, 118, 240, 122, 0, 378, 320, 260, 0, 134, 136

Offset: 1

Michael Somos, Aug 22 2008

			q + 6*q^3 + 4*q^4 + 10*q^5 + 14*q^7 + 16*q^8 + 27*q^9 + 22*q^11 + 24*q^12 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001620, A027748, A038040, A112329, A124010.

a143520 n = product $ zipWith (\p e -> (e + 2 * mod p 2 - 1) * p ^ e)
                              (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jan 21 2014

Abs@Total[# (-1)^Divisors[#]] & /@ Range[68] (* George Beck, Oct 25 2014 *)
f[p_, e_] := (e + 1)*p^e; f[2, e_] := (e - 1)*2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; (e - (-1)^p) * p^e)))}

{a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k * x^k / (1 - (-x)^k)^2, x*O(x^n)), n))}

a(n) is multiplicative with a(2^e) = (e-1) * 2^e if e>0, a(p^e) = (e+1) * p^e if p>2.
a(4*n + 2) = 0.
G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k)^2.
A038040(2*n + 1) = a(2*n + 1); 4 * A038040(n) = a(4*n).
From Amiram Eldar, Nov 29 2022: (Start)
a(n) = n * A112329(n).
Dirichlet g.f.: zeta(s-1)^2*(1+2^(3-2*s)-2^(2-s)).
Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma-1)*n^2/8, where gamma is Euler's constant (A001620). (End)

A094572 Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.

Original entry on oeis.org

2, 0, 4, 2, 4, 0, 4, 4, 6, 0, 4, 4, 4, 0, 8, 6, 4, 0, 4, 4, 8, 0, 4, 8, 6, 0, 8, 4, 4, 0, 4, 8, 8, 0, 8, 6, 4, 0, 8, 8, 4, 0, 4, 4, 12, 0, 4, 12, 6, 0, 8, 4, 4, 0, 8, 8, 8, 0, 4, 8, 4, 0, 12, 10, 8, 0, 4, 4, 8, 0, 4, 12, 4, 0, 12, 4, 8, 0, 4, 12, 10, 0, 4, 8, 8, 0, 8, 8, 4, 0, 8, 4, 8, 0, 8, 16, 4, 0, 12, 6

Offset: 1

N. J. A. Sloane, Nov 02 2008

The old entry with this sequence number was a duplicate of A058071.

a(n) == 2 (mod 4) if n is a square otherwise a(n) is divisible by 4. Cf. A112329. - Peter Bala, Jan 08 2025

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).

Cf. A000005, A001620, A093061, A112329.

with(numtheory); f:=proc(n) if n mod 2 = 1 then RETURN(2*tau(n)); fi; if n mod 4 = 0 then RETURN(2*tau(n/4)); fi; 0; end;

Table[If[OddQ[n],2DivisorSigma[0,n],If[OddQ[n/2],0,2DivisorSigma[0,n/4]]],{n,100}] (* Ray Chandler, Aug 23 2014 *)

a(n) = if(n%2, 2 * numdiv(n), if(n % 4 == 0, 2 * numdiv(n/4), 0)); \\ Amiram Eldar, Apr 13 2024

a(n) = 2*d(n) if n is odd, = 2*d(n/4) if n == 0 mod 4, otherwise 0, where d() = A000005().
a(n) = 2 * A112329(n). - Ray Chandler, Aug 23 2014
From Amiram Eldar, Apr 13 2024: (Start)
Dirichlet g.f.: 2*zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s)).
Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma-1)*n, where gamma is Euler's constant (A001620). (End)

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 16, 28, 47, 78, 126, 198, 306, 464, 694, 1024, 1490, 2146, 3061, 4322, 6052, 8408, 11592, 15872, 21592, 29192, 39242, 52468, 69788, 92376, 121716, 159664, 208569, 271372, 351732, 454228, 584546, 749720, 958472, 1221560, 1552210, 1966698

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305121 and A000009.

The g.f. Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 - x^k) = Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 + x^k - 2*x^k) is congruent mod 2 to Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) = -G(-x^2), where G(x) is the g.f. of A112329. It follows that a(n) is odd iff n = 2*k^2 for some positive integer k. - Peter Bala, Jan 07 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A305102, A305101, A305104, A305105, A305124.

Cf. A116680, A112329, A305121.

nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = A305101(n) - A305124(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A265290 Decimal expansion of Sum_{n>=1} |phi - c(n)|, where phi is the golden ratio (A001622) and c(n) are the convergents to phi.

Original entry on oeis.org

1, 1, 9, 5, 9, 5, 5, 7, 8, 6, 0, 1, 7, 5, 1, 3, 5, 9, 6, 0, 0, 3, 4, 7, 4, 8, 0, 0, 0, 2, 1, 3, 0, 2, 0, 2, 0, 2, 7, 5, 5, 1, 6, 2, 0, 9, 5, 8, 2, 5, 9, 8, 4, 8, 6, 4, 8, 7, 3, 3, 8, 8, 3, 6, 2, 8, 5, 0, 9, 1, 2, 6, 9, 0, 6, 1, 3, 7, 6, 8, 2, 2, 2, 0, 5, 4

Offset: 1

Clark Kimberling, Dec 06 2015

Define the deviance of x > 0 by dev(x) = Sum_{n>=1} |x - c(n,x)|, where c(n,x) = n-th convergent to x. The greatest value of dev(x) occurs when x = golden ratio, so that this constant is the maximal deviance.

			1.195955786017513596003474800021...

Cf. A000045, A001622, A290565.

Cf. A265288, A265289.

x := (3 - sqrt(5))/2:
evalf(sqrt(5)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..16), 100); # Peter Bala, Aug 21 2022

x = GoldenRatio; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265288 *)
RealDigits[s2, 10, 120][[1]]  (* A265289 *)
RealDigits[s1 + s2, 10, 120][[1]] (* A265290, dev(x) *)
d[x_] := If[IntegerQ[1000!*x], Total[Abs[x - Convergents[x]]],
  Total[Abs[x - Convergents[x, 30]]]]
Plot[{d[x], 1.195}, {x, 0, 1}]

Equals Sum_{n>=1} 1/(F(2*n-1)*F(2*n)), where F(n) is the n-th Fibonacci number (A000045).
From Amiram Eldar, Oct 05 2020: (Start)
Equals Sum_{k>=1} 1/(phi^k * F(k)).
Equals sqrt(5) * Sum_{k>=1} 1/(phi^(2*k) - (-1)^k) = sqrt(5) * Sum_{k>=1} (-1)^(k+1)/(phi^(2*k) + (-1)^k).
Equals (A290565 + 1/phi)/2. (End)
A rapidly converging series for the constant is sqrt(5)*Sum_{k >= 1} x^(k^2)*(1 + x^(2*k))/(1 - x^(2*k)), where x = (3 - sqrt(5))/2. See A112329. - Peter Bala, Aug 21 2022

A265293 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

Original entry on oeis.org

5, 1, 7, 1, 7, 4, 2, 2, 0, 2, 2, 0, 6, 7, 1, 8, 8, 6, 2, 1, 9, 9, 6, 4, 3, 5, 2, 3, 3, 8, 6, 6, 9, 2, 3, 6, 1, 0, 5, 5, 2, 1, 3, 5, 7, 3, 4, 9, 9, 7, 1, 0, 5, 3, 5, 4, 7, 1, 9, 1, 6, 6, 3, 7, 3, 7, 1, 8, 9, 8, 5, 8, 8, 2, 3, 3, 0, 3, 0, 8, 5, 2, 9, 6, 5, 8

Offset: 0

Clark Kimberling, Dec 06 2015

			sum = 0.51717422022067188621996435233866923610552...

Cf. A000129, A002193, A112329, A265291, A265292, A265288 (guide).

x := 3 - 2*sqrt(2):
evalf(2*sqrt(2)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..12), 100); # Peter Bala, Aug 20 2022

x = Sqrt[2]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265291 *)
RealDigits[s2, 10, 120][[1]]  (* A265292 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)

From Peter Bala, Aug 20 2022: (Start)
Constant equals Sum_{n >= 1} 1/((1 + sqrt(2))^n*Pell(n)) =  2*sqrt(2)*Sum_{n >= 1} 1/( (3 + 2*sqrt(2))^n - (-1)^n ), where Pell(n) = A000129(n).
A more rapidly converging series for the constant is 2*sqrt(2)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 3 - 2*sqrt(2). See A112329. (End)

cp's OEIS Frontend

A246063 First occurrence of n in sequence A112329.

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A048272 Number of odd divisors of n minus number of even divisors of n.

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A279396 Triangle read by rows T(n, m) = sigma^*(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^*(k)(n) given in a comment in A279395.

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A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

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A143520 a(n) is n times number of divisors of n if n is odd, zero if n is twice odd, n times number of divisors of n/4 if n is divisible by 4.

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A094572 Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.

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A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

A265293 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).