A350123
a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.
Original entry on oeis.org
1, 8, 22, 57, 91, 185, 247, 402, 545, 775, 917, 1379, 1573, 1995, 2455, 3106, 3428, 4377, 4775, 5909, 6753, 7727, 8301, 10331, 11230, 12564, 13904, 15990, 16888, 19908, 20930, 23597, 25545, 27767, 29827, 34468, 35910, 38660, 41328, 46318, 48080, 53644, 55578
Offset: 1
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Accumulate[Table[2*k*DivisorSigma[1, k] - DivisorSigma[2, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)
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a(n) = sum(k=1, n, k^2*(n\k)^2);
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a(n) = sum(k=1, n, k^2*sumdiv(k, d, (2*d-1)/d^2));
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a(n) = sum(k=1, n, 2*k*sigma(k)-sigma(k, 2));
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my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))
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from math import isqrt
def A350123(n): return (-(s:=isqrt(n))**3*(s+1)*((s<<1)+1)+sum((q:=n//k)*(6*k**2*q+((k<<1)-1)*(q+1)*((q<<1)+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 24 2023
A350106
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.
Original entry on oeis.org
1, 1, 4, 1, 6, 8, 1, 10, 14, 15, 1, 18, 32, 31, 21, 1, 34, 86, 87, 45, 33, 1, 66, 248, 295, 153, 81, 41, 1, 130, 734, 1095, 669, 309, 101, 56, 1, 258, 2192, 4231, 3201, 1521, 443, 150, 69, 1, 514, 6566, 16647, 15765, 8373, 2633, 722, 191, 87, 1, 1026, 19688, 66055, 78393, 48321, 17411, 4746, 1005, 253, 99
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
4, 6, 10, 18, 34, 66, 130, ...
8, 14, 32, 86, 248, 734, 2192, ...
15, 31, 87, 295, 1095, 4231, 16647, ...
21, 45, 153, 669, 3201, 15765, 78393, ...
33, 81, 309, 1521, 8373, 48321, 284709, ...
41, 101, 443, 2633, 17411, 119321, 828323, ...
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T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 14 2021 *)
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T(n, k) = sum(j=1, n, j*(n\j)^k);
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T(n, k) = sum(j=1, n, j*sumdiv(j, d, (d^k-(d-1)^k)/d));
A350128
a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.
Original entry on oeis.org
1, 8, 44, 417, 4545, 69905, 1207937, 24904806, 575256641, 14947281595, 427836523971, 13429362462839, 457637290140469, 16843379604615375, 665494379869134005, 28102480944522059434, 1262906802939553227382, 60182948301301262753877
Offset: 1
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Table[Sum[k^n Floor[n/k]^2,{k,n}],{n,20}] (* Harvey P. Dale, Feb 11 2022 *)
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a(n) = sum(k=1, n, k^n*(n\k)^2);
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a(n) = sum(k=1, n, 2*k*sigma(k, n-1)-sigma(k, n));
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from math import isqrt
from sympy import bernoulli
def A350128(n): return (((s:=isqrt(n))+1)*(1-s)*(bernoulli(n+1,s+1)-(b:=bernoulli(n+1)))+sum(k**n*(n+1)*(((q:=n//k)+1)*(q-1))+(1-2*k)*(b-bernoulli(n+1,q+1)) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023
Showing 1-3 of 3 results.