A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).
1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29
Offset: 1
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 3, 4, 6, 10, 18, 34, ... 5, 8, 16, 38, 100, 278, ... 8, 15, 37, 111, 373, 1335, ... 10, 21, 63, 237, 999, 4461, ... 14, 33, 113, 489, 2393, 12513, ...
Crossrefs
Programs
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Mathematica
Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten -
Python
from itertools import count, islice from math import isqrt from sympy import bernoulli def A319649_T(n,k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1,q+1) for w in range(1,s+1)))//(k+1) + int(k==0) def A319649_gen(): # generator of terms return (A319649_T(k+1,n-k-1) for n in count(1) for k in range(n)) A319649_list = list(islice(A319649_gen(),30)) # Chai Wah Wu, Oct 24 2023
Formula
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} j^k*x^j/(1 - x^j).
A(n,k) = Sum_{j=1..n} sigma_k(j).