A224923 -id:A224923 - cp's OEIS frontend

A072481 a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d).

0, 0, 0, 1, 2, 6, 9, 17, 25, 37, 50, 72, 89, 117, 148, 184, 220, 271, 318, 382, 443, 513, 590, 688, 773, 876, 988, 1113, 1237, 1388, 1526, 1693, 1860, 2044, 2241, 2459, 2657, 2890, 3138, 3407, 3665, 3962, 4246, 4571, 4899, 5238, 5596, 5999, 6373, 6787, 7207

Offset: 0

Reinhard Zumkeller, Aug 02 2002

nonn

Previous name was: Sums of sums of remainders when dividing n by k, 0

Partial sums of A004125.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A224923, A224924.

N:= 200: # to get a(0) to a(N)
S:= series(add(k*x^(2*k)/(1-x^k),k=1..floor(N/2))/(1-x)^2, x, N+1):
seq((n^3-n)/6 - coeff(S,x,n), n=0..N); # Robert Israel, Aug 13 2015

a[n_] := n(n+1)(2n+1)/6 - Sum[DivisorSigma[1, k] (n-k+1), {k, 1, n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 08 2019, after Omar E. Pol *)

a(n) = sum(k=1, n, sum(d=1, k, k % d)); \\ Michel Marcus, Feb 11 2014

for n in range(99):
    s = 0
    for k in range(1,n+1):
      for d in range(1,k+1):
        s += k % d
    print(str(s), end=',')

from math import isqrt
def A072481(n): return (n*(n+1)*((n<<1)+1)-((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1)-sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 22 2023

a(n) = Sum_{k=1..n} Sum_{d=1..k}(k mod d).
a(n) = A000330(n) - A175254(n), n >= 1. - Omar E. Pol, Aug 12 2015
G.f.: x^2/(1-x)^4 - (1-x)^(-2) * Sum_{k>=1} k*x^(2*k)/(1-x^k). - Robert Israel, Aug 13 2015
a(n) ~ (1 - Pi^2/12)*n^3/3. - Vaclav Kotesovec, Sep 25 2016

New name and a(0) from Alex Ratushnyak, Feb 10 2014

A224924 Sum_{i=0..n} Sum_{j=0..n} (i AND j), where AND is the binary logical AND operator.

Original entry on oeis.org

0, 1, 3, 12, 16, 33, 63, 112, 120, 153, 211, 300, 408, 553, 735, 960, 976, 1041, 1155, 1324, 1536, 1809, 2143, 2544, 2952, 3433, 3987, 4620, 5320, 6105, 6975, 7936, 7968, 8097, 8323, 8652, 9072, 9601, 10239, 10992, 11800, 12729, 13779, 14956, 16248, 17673, 19231, 20928

Offset: 0

Alex Ratushnyak, Apr 19 2013

For n>0, a(2^n)-A000217(2^n)=a(2^n-1)-A000217(2^n-1) [See links]. - R. J. Cano, Aug 21 2013

Enrique Pérez Herrero, Table of n, a(n) for n = 0..1000
R. J. Cano, Additional information
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.

Cf. A004125, A222423, A224915, A224923, A000217.

read("transforms") :
A224924 := proc(n)
    local a,i,j ;
    a := 0 ;
    for i from 0 to n do
    for j from 0 to n do
        a := a+ANDnos(i,j) ;
    end do:
    end do:
    a ;
end proc: # R. J. Mathar, Aug 22 2013

a[n_] := Sum[BitAnd[i, j], {i, 0, n}, {j, 0, n}];
Table[a[n], {n, 0, 20}]
(* Enrique Pérez Herrero, May 30 2015 *)

a(n)=sum(i=0,n,sum(j=0,n,bitand(i,j))); \\ R. J. Cano, Aug 21 2013

for n in range(99):
    s = 0
    for i in range(n+1):
      for j in range(n+1):
        s += i & j
    print(s, end=',')

a(2^n) = a(2^n - 1) + 2^n.

a(n) -a(n-1) = 2*A222423(n) -n. - R. J. Mathar, Aug 22 2013

A224932 Maximal side length b <= a = n of integer parallelograms.

Original entry on oeis.org

0, 0, 0, 3, 5, 0, 6, 6, 8, 10, 10, 11, 13, 13, 15, 15, 17, 16, 18, 20, 20, 21, 21, 23, 25, 26, 26, 27, 29, 30, 29, 31, 31, 34, 35, 33, 37, 37, 39, 40, 41, 41, 41, 43, 45, 45, 46, 46, 48, 50

Offset: 1

Reiner Moewald, Apr 20 2013

nonn

Reiner Moewald, Table of n, a(n) for n = 1..100

Cf. A224921, A224923.

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A072481 a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d).

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A224924 Sum_{i=0..n} Sum_{j=0..n} (i AND j), where AND is the binary logical AND operator.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A224932 Maximal side length b <= a = n of integer parallelograms.

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Author

Keywords

Links

Crossrefs