A049766 -id:A049766 - cp's OEIS frontend

A049765 Triangular array T, read by rows: T(n,k) = (k mod n) + (n mod k), for k = 1..n and n >= 1.

0, 1, 0, 1, 3, 0, 1, 2, 4, 0, 1, 3, 5, 5, 0, 1, 2, 3, 6, 6, 0, 1, 3, 4, 7, 7, 7, 0, 1, 2, 5, 4, 8, 8, 8, 0, 1, 3, 3, 5, 9, 9, 9, 9, 0, 1, 2, 4, 6, 5, 10, 10, 10, 10, 0, 1, 3, 5, 7, 6, 11, 11, 11, 11, 11, 0, 1, 2, 3, 4, 7, 6, 12, 12, 12, 12, 12, 0

Offset: 1

Clark Kimberling

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  0;
  1, 0;
  1, 3, 0;
  1, 2, 4, 0;
  1, 3, 5, 5, 0;
  1, 2, 3, 6, 6,  0;
  1, 3, 4, 7, 7,  7,  0;
  1, 2, 5, 4, 8,  8,  8,  0;
  1, 3, 3, 5, 9,  9,  9,  9,  0;
  1, 2, 4, 6, 5, 10, 10, 10, 10, 0;
  ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Row sums are in A049766.

Cf. A048158, A049767, A049768.

Flat(List([1..15], n-> List([1..n], k-> (k mod n) + (n mod k) ))); # G. C. Greubel, Dec 13 2019

[[(k mod n) + (n mod k): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Dec 13 2019

seq(seq( `mod`(k, n) + `mod`(n, k), k = 1..n), n = 1..15); # G. C. Greubel, Dec 13 2019

Table[Mod[k,n] + Mod[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Dec 13 2019 *)

T(n,k) = k%n + n%k;
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 13 2019

[[(k%n) + (n%k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 13 2019

A367379 a(n) = Sum_{j=1..n} Sum_{i=1..n} (j mod i).

Original entry on oeis.org

0, 1, 5, 12, 26, 44, 73, 109, 157, 215, 292, 375, 481, 603, 744, 900, 1087, 1287, 1522, 1773, 2053, 2361, 2712, 3073, 3476, 3913, 4389, 4891, 5448, 6021, 6653, 7316, 8028, 8786, 9599, 10427, 11326, 12277, 13287, 14325, 15442, 16587, 17815, 19089, 20418, 21811

Offset: 1

Darío Clavijo, Nov 15 2023

nonn

Partial sums of A049766.

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

Cf. A000292, A002411, A049766, A072481, A175254.

N:= 100: # for a(1)..a(N)
S:= [seq(NumberTheory:-SumOfDivisors(i,1),i=1..N+1)]:
SS:= ListTools:-PartialSums(S):
S2:= [seq(i*S[i],i=1..N+1)]:
SS2:= ListTools:-PartialSums(S2):
f:= n -> 1/2*n^2*(n+1) - (n+1)*SS[n+1]+SS2[n+1]:
map(f, [$1..N]); # Robert Israel, Dec 20 2023

a[n_]:=n^2(n+1)/2-Sum[DivisorSigma[1,i](n-i+1),{i,n+1}]; Array[a,47,0] (* Stefano Spezia, Nov 17 2023 *)

a(n) = sum(j=1, n, sum(i=1, n, j % i)); \\ Michel Marcus, Nov 16 2023

from sympy import divisor_sigma
A002411 = lambda n: ((n*n)*(n+1))>>1
A175254 = lambda n: sum(divisor_sigma(i) * (n-i+1) for i in range(1,n+1))
a = lambda n: A002411(n) - A175254(n)
print([a(n) for n in range(1, 53)])

from math import isqrt
def A367379(n): return (n**2*(n+1)>>1)-(((s:=isqrt(n))**2*(s+1)*((s+1)*(2*s+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+2*q+1)+3*(n+1)*(2*k+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Dec 20 2023

a(n) = A072481(n) + A000292(n-1).

a(n) = A002411(n) - A175254(n).

cp's OEIS Frontend

A049765 Triangular array T, read by rows: T(n,k) = (k mod n) + (n mod k), for k = 1..n and n >= 1.

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