A024919 -id:A024919 - cp's OEIS frontend

A002129 Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.

1, -1, 4, -5, 6, -4, 8, -13, 13, -6, 12, -20, 14, -8, 24, -29, 18, -13, 20, -30, 32, -12, 24, -52, 31, -14, 40, -40, 30, -24, 32, -61, 48, -18, 48, -65, 38, -20, 56, -78, 42, -32, 44, -60, 78, -24, 48, -116, 57, -31, 72, -70, 54, -40, 72, -104, 80, -30, 60, -120, 62, -32, 104, -125

Offset: 1

N. J. A. Sloane

Glaisher calls this zeta(n) or zeta_1(n). - N. J. A. Sloane, Nov 24 2018
Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n)^2 = Sum_{n >= 1} (-1)^(n-1)*n*x^n/(1-x^n).
Unsigned sequence is A113184. - Peter Bala, Dec 14 2020

			a(28) = 40 because the sum of the even divisors of 28 (2, 4, 14 and 28) = 48 and the sum of the odd divisors of 28 (1 and 7) = 8, their absolute difference being 40.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 3rd formula.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Steven R. Finch, The "One-Ninth" Constant [Broken link]
Steven R. Finch, The "One-Ninth" Constant [From the Wayback machine]
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).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
Index entries for sequences mentioned by Glaisher

A diagonal of A060044.
a(2^n) = -A036563(n+1). a(3^n) = A003462(n+1).
First differences of -A024919(n).
Cf. A010054, A113184.

A002129 := proc(n) -add((-1)^d*d,d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 05 2011

f[n_] := Block[{c = Divisors@ n}, Plus @@ Select[c, EvenQ] - Plus @@ Select[c, OddQ]]; Array[f, 64] (* Robert G. Wilson v, Mar 04 2011 *)
a[n_] := DivisorSum[n, -(-1)^#*#&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := If[p == 2, 3 - 2^(e + 1), (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]);  Array[a, 64] (* Amiram Eldar, Jul 20 2019 *)

PARI
```
a(n)=if(n<1,0,-sumdiv(n,d,(-1)^d*d))
```

{a(n)=n*polcoeff(log(sum(k=0,(sqrtint(8*n+1)-1)\2,x^(k*(k+1)/2))+x*O(x^n)),n)} \\ Paul D. Hanna, Jun 28 2008

Multiplicative with a(p^e) = 3-2^(e+1) if p = 2; (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Sep 01 2001
G.f.: Sum_{n>=1} n*x^n*(1-3*x^n)/(1-x^(2*n)). - Vladeta Jovovic, Oct 15 2002
L.g.f.: Sum_{n>=1} a(n)*x^n/n = log[ Sum_{n>=0} x^(n(n+1)/2) ], the log of the g.f. of A010054. - Paul D. Hanna, Jun 28 2008
Dirichlet g.f. zeta(s)*zeta(s-1)*(1-4/2^s). Dirichlet convolution of A000203 and the quasi-finite (1,-4,0,0,0,...). - R. J. Mathar, Mar 04 2011
a(n) = A000593(n)-A146076(n). - R. J. Mathar, Mar 05 2011
a(n) = Sum_{j = 1..n} Sum_{k = 1..j} (-1)^(j+1)*cos(2*k*n*Pi/j). - Peter Bala, Aug 24 2022
G.f.: Sum_{n>=1} n*(-x)^(n-1)/(1-x^n). - Mamuka Jibladze, Jun 03 2025

Better description and more terms from Robert G. Wilson v, Dec 14 2000

More terms from N. J. A. Sloane, Mar 19 2001

A366915 a(n) = Sum_{k=1..n} (-1)^kk^2floor(n/k).

Original entry on oeis.org

-1, 2, -8, 11, -15, 15, -35, 48, -43, 35, -87, 103, -67, 83, -177, 162, -128, 145, -217, 277, -223, 143, -387, 443, -208, 302, -518, 432, -410, 370, -592, 771, -449, 421, -879, 850, -520, 566, -1134, 1024, -658, 842, -1008, 1310, -1056, 534, -1676, 1714, -737

Offset: 1

Chai Wah Wu, Oct 28 2023

sign

Cf. A024919, A064602.

a[n_]:=Sum[ (-1)^k*k^2*Floor[n/k],{k,n}]; Array[a,49] (* Stefano Spezia, Oct 29 2023 *)

a(n) = sum(k=1, n, (-1)^k*k^2*(n\k)); \\ Michel Marcus, Oct 29 2023

from math import isqrt
def A366915(n): return (-(t:=isqrt(m:=n>>1))**2*(t+1)*((t<<1)+1)+sum((q:=m//k)*(6*k**2+q*((q<<1)+3)+1) for k in range(1,t+1))<<2)//3+((s:=isqrt(n))**2*(s+1)*((s<<1)+1)-sum((q:=n//k)*(6*k**2+q*((q<<1)+3)+1) for k in range(1,s+1)))//6

a(n) = 8*A064602(floor(n/2))-A064602(n).

A366917 a(n) = Sum_{k=1..n} (-1)^kk^3floor(n/k).

Original entry on oeis.org

-1, 6, -22, 49, -77, 119, -225, 358, -399, 483, -849, 1139, -1059, 1349, -2179, 2500, -2414, 2885, -3975, 4971, -4661, 4663, -7505, 8819, -6932, 8454, -11986, 12438, -11952, 12744, -17048, 20399, -16897, 17501, -25843, 27904, -22750, 25270, -36274, 37184, -31738

Offset: 1

Chai Wah Wu, Oct 28 2023

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A024919, A366915, A064603.

f:= proc(n) local k; add((-1)^k * k^3 * floor(n/k), k=1..n) end proc;
map(f, [$1..100]); # Robert Israel, Dec 29 2023

a[n_]:=Sum[ (-1)^k*k^3*Floor[n/k],{k,n}]; Array[a,41] (* Stefano Spezia, Oct 29 2023 *)

a(n) = sum(k=1, n, (-1)^k*k^3*(n\k)); \\ Michel Marcus, Oct 29 2023

from math import isqrt
def A366917(n): return (-(t:=isqrt(m:=n>>1))**3*(t+1)**2+sum((q:=m//k)*((k**3<<2)+q*(q*(q+2)+1)) for k in range(1,t+1))<<2)+((s:=isqrt(n))**3*(s+1)**2 - sum((q:=n//k)*((k**3<<2)+q*(q*(q+2)+1)) for k in range(1,s+1))>>2)

a(n) = 16*A064603(floor(n/2)) - A064603(n).

A366937 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+1,2) * floor(n/k).

Original entry on oeis.org

1, -1, 6, -6, 10, -7, 22, -26, 26, -16, 51, -54, 38, -41, 101, -83, 71, -72, 119, -143, 123, -66, 211, -230, 111, -151, 279, -216, 220, -182, 315, -397, 237, -207, 467, -430, 274, -279, 599, -519, 343, -423, 524, -665, 557, -250, 879, -874, 380, -612, 874, -776

Offset: 1

Seiichi Manyama, Oct 29 2023

sign

Partial sums of A320900.
Cf. A024919, A366938, A366939.
Cf. A364970, A366395.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+1, 2)*(n\k));

from math import isqrt
def A366937(n): return (((s:=isqrt(m:=n>>1))*(s+1)**2*((s<<2)+5)<<1)-(t:=isqrt(n))*(t+1)**2*(t+2)-sum((((q:=m//w)+1)*(q*((q<<2)+5)+6*w*((w<<1)+1))<<1) for w in range(1,s+1))+sum(((q:=n//w)+1)*(q*(q+2)+3*w*(w+1)) for w in range(1,t+1)))//6 # Chai Wah Wu, Oct 29 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^3 = -1/(1-x) * Sum_{k>=1} binomial(k+1,2) * (-x)^k/(1-x^k).

A366938 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+2,3) * floor(n/k).

Original entry on oeis.org

1, -2, 9, -14, 22, -27, 58, -85, 91, -97, 190, -243, 213, -266, 460, -499, 471, -553, 778, -970, 896, -845, 1456, -1697, 1264, -1560, 2270, -2289, 2207, -2307, 3150, -3793, 3049, -3125, 4765, -5079, 4061, -4492, 6634, -6714, 5628, -6370, 7821, -9120, 7986, -7013

Offset: 1

Seiichi Manyama, Oct 29 2023

sign

Partial sums of A320901.
Cf. A024919, A366937, A366939.
Cf. A365409, A366659.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+2, 3)*(n\k));

from math import isqrt
def A366938(n): return (((s:=isqrt(m:=n>>1))*(s+1)**3*(s+2)<<4)-(t:=isqrt(n))*(t+1)**2*(t+2)*(t+3)-sum((((q:=m//w)+1)*(q*(q+1)*(q+2)+(w*(w+1)*((w<<1)+1)<<1))<<4) for w in range(1,s+1))+sum(((q:=n//w)+1)*(q*(q+2)*(q+3)+(w*(w+1)*(w+2)<<2)) for w in range(1,t+1)))//24 # Chai Wah Wu, Oct 29 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^4 = -1/(1-x) * Sum_{k>=1} binomial(k+2,3) * (-x)^k/(1-x^k).

A366939 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+3,4) * floor(n/k).

Original entry on oeis.org

1, -3, 13, -26, 45, -70, 141, -228, 283, -366, 636, -879, 942, -1232, 1914, -2331, 2515, -3090, 4226, -5313, 5539, -6114, 8837, -10558, 9988, -11947, 15969, -17705, 18256, -20364, 26013, -30592, 29330, -31874, 42222, -47034, 44357, -49602, 64164, -69115, 66637, -74017

Offset: 1

Seiichi Manyama, Oct 29 2023

sign

Cf. A024919, A366937, A366938.

Cf. A365439, A366723.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+3, 4)*(n\k));

from math import isqrt
from sympy import rf
def A366939(n): return ((rf(s:=isqrt(m:=n>>1),3)*(s+1)*((s**2<<2)+13*s+8)<<3)-rf(t:=isqrt(n),5)*(t+1)+sum((((q:=m//w)+1)*(-q*(q+2)*((q**2<<2)+13*q+8)-5*w*(w+1)*((r:=w<<1)+1)*(r+3))<<3) for w in range(1,s+1))+sum(rf(q:=n//w,5)+5*(q+1)*rf(w,4) for w in range(1,t+1)))//120 # Chai Wah Wu, Oct 29 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^5 = -1/(1-x) * Sum_{k>=1} binomial(k+3,4) * (-x)^k/(1-x^k).

A309124 a(n) = n - 3 * floor(n/3) + 5 * floor(n/5) - 7 * floor(n/7) + ...

Original entry on oeis.org

1, 2, 0, 1, 7, 5, -1, 0, 7, 13, 3, 1, 15, 9, -3, -2, 16, 23, 5, 11, 23, 13, -9, -11, 20, 34, 14, 8, 38, 26, -4, -3, 17, 35, -1, 6, 44, 26, -2, 4, 46, 58, 16, 6, 48, 26, -20, -22, 21, 52, 16, 30, 84, 64, 4, -2, 34, 64, 6, -6, 56, 26, -16, -15, 69, 89, 23, 41, 85, 49, -21, -14, 60, 98, 36

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

sign

Partial sums of A050457.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014200, A024919, A050457, A078471.

f:= proc(n) local r,d;
  r:= n/2^padic:-ordp(n,2);
  add((-1)^((d-1)/2)*d, d = numtheory:-divisors(r))
end proc:
ListTools:-PartialSums(map(f,[$1..100])); # Robert Israel, Oct 28 2020

Table[Sum[(-1)^(k + 1) (2 k - 1) Floor[n/(2 k - 1)], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) (2 k - 1) x^(2 k - 1)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

A366919 a(n) = Sum_{k=1..n} (-1)^kk^nfloor(n/k).

Original entry on oeis.org

-1, 2, -22, 203, -2285, 33855, -609345, 12420372, -284964519, 7347342215, -209807114169, 6554034238459, -222469737401739, 8159109186320903, -321461264348047819, 13538455640979049698, -606976994365011212414, 28864017965496692865925, -1451086990386146504580735

Offset: 1

Chai Wah Wu, Oct 28 2023

sign

Cf. A024919, A308313, A366915, A366917, A319649.

a[n_]:=Sum[ (-1)^k*k^n*Floor[n/k],{k,n}]; Array[a,19] (* Stefano Spezia, Oct 29 2023 *)

a(n) = sum(k=1, n, (-1)^k*k^n*(n\k)); \\ Michel Marcus, Oct 29 2023

from math import isqrt
from sympy import bernoulli
def A366919(n): return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<

a(n) = (-1)^n*A308313(n).

Let A(n,k) = Sum_{j=1..n} j^k * floor(n/j). Then a(n) = 2^(n+1)*A(floor(n/2),n)-A(n,n).

A072663 Numbers m such that Sum_{k=1..m} (-1)^kkfloor(m/k) = 0.

Original entry on oeis.org

2, 26, 28, 76, 210, 1801, 3508, 16180, 29286, 33988, 1161208, 4010473, 164048770, 18294479654

Offset: 1

Benoit Cloitre, Aug 10 2002

It is easy to see that if f(n) = A024919(n) = Sum_{k=1..n} (-1)^k*k*floor(n/k) then f(n) = f(n-1) + (2^(L+1)-3)*sigma(M) if n=2^L*M, where M is odd and L >= 0. Using this we can get a faster program to calculate this sequence. - Robert Gerbicz, Aug 30 2002

The zeros of A024919.

f[n_] := Sum[(-1)^i*i*Floor[n/i], {i, 1, n}]; Do[s = f[n]; If[s == 0, Print[n]], {n, 1, 40000}]

lista(nn) = {my(s=-1); for(m=2, nn, x=bitand(m, -m); if((s+=(2*x-3)*sigma(m/x)) == 0, print1(m, ", "))); } \\ Jinyuan Wang, Apr 06 2020

Four more terms from Klaus Brockhaus, Aug 13 2002
More terms from Robert Gerbicz, Aug 30 2002
a(14) from Giovanni Resta, Apr 06 2020

A333505 a(n) = Sum_{k=1..n} (-1)^(k+1) * k * ceiling(n/k).

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 5, 5, 1, 4, 9, 9, 2, 2, 9, 17, 5, 5, 11, 11, 2, 12, 23, 23, -4, 1, 14, 26, 15, 15, 22, 22, -6, 8, 25, 37, 9, 9, 28, 44, 7, 7, 18, 18, 3, 35, 58, 58, -9, -2, 18, 38, 21, 21, 36, 52, 5, 27, 56, 56, -3, -3, 28, 68, 8, 26, 45, 45, 24, 50, 73, 73, -23, -23, 14

Offset: 1

Ilya Gutkovskiy, May 25 2020

sign

Cf. A001057, A002129, A006590, A024916, A024919, A332490, A332682.

Table[Sum[(-1)^(k + 1) k Ceiling[n/k], {k, 1, n}], {n, 1, 75}]
Table[(-1)^(n + 1) Ceiling[n/2] + Sum[DivisorSum[k, (-1)^(# + 1) # &], {k, 1, n - 1}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[x/(1 - x) (1/(1 + x)^2 + Sum[(-1)^(k + 1) k x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (-1)^(k+1)*k*ceil(n/k)); \\ Michel Marcus, May 26 2020

from math import isqrt
def A333505(n): return ((s:=isqrt(m:=n-1>>1))**2*(s+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))<<1)-((t:=isqrt(n-1))**2*(t+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,t+1))>>1) + (m+1 if n&1 else -m-1) # Chai Wah Wu, Oct 30 2023

G.f.: (x/(1 - x)) * (1/(1 + x)^2 + Sum_{k>=1} (-1)^(k+1) * k * x^k / (1 - x^k)).
a(n) = (-1)^(n+1) * ceiling(n/2) + Sum_{k=1..n-1} A002129(k).
a(n) = A001057(n) - A024919(n-1).

cp's OEIS Frontend

A002129 Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.

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A366915 a(n) = Sum_{k=1..n} (-1)^k*k^2*floor(n/k).

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A366917 a(n) = Sum_{k=1..n} (-1)^k*k^3*floor(n/k).

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A366937 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+1,2) * floor(n/k).

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A366938 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+2,3) * floor(n/k).

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A366939 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+3,4) * floor(n/k).

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A309124 a(n) = n - 3 * floor(n/3) + 5 * floor(n/5) - 7 * floor(n/7) + ...

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A366919 a(n) = Sum_{k=1..n} (-1)^k*k^n*floor(n/k).

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A366915 a(n) = Sum_{k=1..n} (-1)^kk^2floor(n/k).

A366917 a(n) = Sum_{k=1..n} (-1)^kk^3floor(n/k).

A366919 a(n) = Sum_{k=1..n} (-1)^kk^nfloor(n/k).

A072663 Numbers m such that Sum_{k=1..m} (-1)^kkfloor(m/k) = 0.