A059851 -id:A059851 - cp's OEIS frontend

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

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

A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

Original entry on oeis.org

-1, 1, -1, 2, 0, 4, 2, 6, 3, 7, 5, 11, 9, 13, 9, 14, 12, 18, 16, 22, 18, 22, 20, 28, 25, 29, 25, 31, 29, 37, 35, 41, 37, 41, 37, 46, 44, 48, 44, 52, 50, 58, 56, 62, 56, 60, 58, 68, 65, 71, 67, 73, 71, 79, 75, 83, 79, 83, 81, 93, 91, 95, 89, 96, 92, 100, 98, 104, 100, 108

Offset: 1

Ilya Gutkovskiy, Apr 22 2019

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000005, A006218, A051950, A059851, A068762, A068773, A092405, A263084 (bisection).

Cf. A001620 (gamma), A002162.

nmax = 70; Rest[CoefficientList[Series[1/(1 - x) Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^k DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
Accumulate[Array[(-1)^#*DivisorSigma[0, #] &, 70]] (* Amiram Eldar, Oct 14 2022 *)

a(n) = Sum_{k=1..n} (-1)^k*A000005(k).

a(n) = n*log(n)/2 + (gamma - log(2) - 1/2)*n + O(n^(131/416 + eps)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022

A268235 a(n) = Sum_{k=1..n} floor(n/k)*2^(k-1).

Original entry on oeis.org

1, 4, 9, 20, 37, 76, 141, 280, 541, 1072, 2097, 4192, 8289, 16548, 32953, 65860, 131397, 262764, 524909, 1049736, 2098381, 4196560, 8390865, 16781696, 33558929, 67117460, 134226585, 268452580, 536888037, 1073775900, 2147517725, 4295034280, 8590002605, 17180002736, 34359872001, 68719743792

Offset: 1

Benoit Cloitre and N. J. A. Sloane, Feb 05 2016

nonn

This is the "floor transform" of the powers of 2.

Matthew House, Table of n, a(n) for n = 1..3305

First differences give A034729.
Cf. A000079.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), this sequence (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A268235:= func< n | (&+[Floor(n/j)*2^(j-1): j in [1..n]]) >;
[A268235(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

# floor transform of a sequence
ft:=proc(a) local b,n,j,k; b:=[];
for n from 1 to nops(a) do j:=add(a[k]*floor(n/k),k=1..n); b:=[op(b),j]; od;
b; end:
ft([seq(2^i,i=0..50)]);

Table[Sum[Floor[n/k] 2^(k - 1), {k, n}], {n, 36}] (* Michael De Vlieger, Feb 12 2017 *)

a(n) = sum(k=1, n, (n\k)*2^(k-1)); \\ Michel Marcus, Feb 11 2017

a(n) = sum(k=1, n, sumdiv(k, d, 2^(d-1))); \\ Seiichi Manyama, May 29 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*x^k))/(1-x)) \\ Seiichi Manyama, May 29 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 2^(k-1)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, May 29 2021

def A268235(n): return sum((n//j)*2^(j-1) for j in range(1,n+1))
[A268235(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) ~ 2^n. - Vaclav Kotesovec, May 28 2021
From Seiichi Manyama, May 29 2021: (Start)
a(n) = Sum_{k=1..n} Sum_{d|k} 2^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 2^(k-1) * x^k/(1 - x^k). (End)
a(n) = Sum_{k=1..n} (2^floor(n/k) - 1). - Ridouane Oudra, Feb 03 2023

Definition corrected by Matthew House, Feb 11 2017

A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A024916, A268235, A308313, A308814, A319194, A320095.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]

a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020

a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021

def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

A344814 a(n) = Sum_{k=1..n} floor(n/k) * 3^(k-1).

Original entry on oeis.org

1, 5, 15, 46, 128, 384, 1114, 3332, 9903, 29671, 88721, 266151, 797593, 2392649, 7175709, 21526834, 64573556, 193720536, 581141026, 1743422288, 5230207428, 15690619684, 47071679294, 141215037738, 423644574301, 1270933715189, 3812799550089, 11438398630159

Offset: 1

Seiichi Manyama, May 29 2021

nonn

Partial sums of A034730.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=3 of A344821.

Cf. A059851, A268235, A332533, A344815, A344816, A344817, A344818, A344819, A344820.

A344814:= func< n | (&+[Floor(n/k)*3^(k-1): k in [1..n]]) >;
[A344814(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(3^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 05 2023

a[n_] := Sum[3^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*3^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, 3^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-3*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 3^(k-1)*x^k/(1-x^k))/(1-x))

def A344814(n): return sum((n//k)*3^(k-1) for k in range(1,n+1))
[A344814(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} 3^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 3*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 3^(k-1) * x^k/(1 - x^k).
a(n) ~ 3^n / 2. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/2) * Sum_{k=1..n} (3^floor(n/k) - 1). - Ridouane Oudra, Feb 05 2023

A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).

Original entry on oeis.org

1, 6, 23, 92, 349, 1394, 5491, 21944, 87497, 349902, 1398479, 5593892, 22371109, 89484074, 357919803, 1431678080, 5726645377, 22906581142, 91626057879, 366504227292, 1466015859181, 5864063418866, 23456249463283, 93824997852744, 375299974563657, 1501199898183502

Offset: 1

Seiichi Manyama, May 29 2021

nonn

Partial sums of A339684.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=4 of A344821.

A344815:= func< n | (&+[Floor(n/j)*4^(j-1): j in [1..n]]) >;
[A344815(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

seq(add(4^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 16 2023

a[n_] := Sum[4^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*4^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, 4^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-x^k))/(1-x))

def A344815(n): return sum((n//j)*4^(j-1) for j in range(1,n+1))
[A344815(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) = Sum_{k=1..n} Sum_{d|k} 4^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 4*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 4^(k-1) * x^k/(1 - x^k).
a(n) ~ 4^n / 3. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/3) * Sum_{k=1..n} (4^floor(n/k) - 1). - Ridouane Oudra, Feb 16 2023

A344816 a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).

Original entry on oeis.org

1, 7, 33, 164, 790, 3946, 19572, 97828, 488479, 2442235, 12207861, 61039267, 305179893, 1525898649, 7629414925, 38147071306, 190734961932, 953674808838, 4768372074464, 23841860356470, 119209292012746, 596046459981502, 2980232250997128, 14901161254984784

Offset: 1

Seiichi Manyama, May 29 2021

nonn

Partial sums of A339685.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=5 of A344821.

A344816:= func< n | (&+[Floor(n/k)*5^(k-1): k in [1..n]]) >;
[A344816(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(5^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

a[n_] := Sum[5^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*5^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, 5^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-5*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 5^(k-1)*x^k/(1-x^k))/(1-x))

def A344816(n): return sum((n//k)*5^(k-1) for k in range(1,n+1))
[A344816(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} 5^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 5*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 5^(k-1) * x^k/(1 - x^k).
a(n) ~ 5^n / 4. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/4) * Sum_{k=1..n} (5^floor(n/k) - 1). - Ridouane Oudra, Mar 05 2023

A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).

Original entry on oeis.org

1, 0, 5, -4, 13, -16, 49, -88, 173, -324, 701, -1384, 2713, -5416, 10989, -21916, 43621, -87224, 174921, -349872, 698773, -1397356, 2796949, -5593872, 11183361, -22366976, 44742149, -89483716, 178951741, -357903312, 715838513, -1431678040, 2863290285

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=2 of A344824.
Cf. A081295.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), this sequence (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344817:= func< n | (&+[Floor(n/k)*(-2)^(k-1): k in [1..n]]) >;
[A344817(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-2)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-2)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-2)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+2*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-2)^(k-1)*x^k/(1-x^k))/(1-x))

def A344817(n): return sum((n//k)*(-2)^(k-1) for k in range(1,n+1))
[A344817(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-2)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-2)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 2^n / 3. - Vaclav Kotesovec, Jun 05 2021

A344820 a(n) = Sum_{k=1..n} floor(n/k) * (-n)^(k-1).

Original entry on oeis.org

1, 0, 9, -52, 520, -6636, 102984, -1864600, 38741463, -909081740, 23775986069, -685854111804, 21633935838489, -740800448012044, 27368368159530285, -1085102592823737200, 45957792326631241516, -2070863582899905915336, 98920982783031811482920, -4993219047619535240997780

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..387

Diagonal of A344824.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): this sequence (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344820:= func< n | (&+[Floor(n/k)*(-n)^(k-1): k in [1..n]]) >;
[A344820(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-n)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 20] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-n)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-n)^(d-1)));

def A344820(n): return sum((n//k)*(-n)^(k-1) for k in range(1,n+1))
[A344820(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-n)^(d-1).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k/(1 + n*x^k).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (-n)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * n^(n-1). - Vaclav Kotesovec, Jun 05 2021

A344818 a(n) = Sum_{k=1..n} floor(n/k) * (-3)^(k-1).

Original entry on oeis.org

1, -1, 9, -20, 62, -174, 556, -1660, 4911, -14693, 44357, -133053, 398389, -1195207, 3587853, -10763270, 32283452, -96850386, 290570104, -871710994, 2615074146, -7845220010, 23535839600, -70607518824, 211822017739, -635466060265, 1906399774635, -5719199303975

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=3 of A344824.
Cf. A101561.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), this sequence (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344818:= func< n | (&+[Floor(n/k)*(-3)^(k-1): k in [1..n]]) >;
[A344818(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-3)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-3)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-3)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+3*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-3)^(k-1)*x^k/(1-x^k))/(1-x))

def A344818(n): return sum((n//k)*(-3)^(k-1) for k in range(1,n+1))
[A344818(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-3)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 3*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-3)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 3^n / 4. - Vaclav Kotesovec, Jun 05 2021

cp's OEIS Frontend

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

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A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

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A268235 a(n) = Sum_{k=1..n} floor(n/k)*2^(k-1).

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A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

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A344814 a(n) = Sum_{k=1..n} floor(n/k) * 3^(k-1).

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