A350109
a(n) = Sum_{k=1..n} k * floor(n/k)^n.
Original entry on oeis.org
1, 6, 32, 295, 3201, 48321, 828323, 16910106, 388005909, 10019717653, 285409876785, 8920506515453, 302901435774351, 11113364096436947, 437903477186179875, 18447307498823123948, 827244767844150424228, 39346708569526147402819
Offset: 1
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a[n_] := Sum[k * Floor[n/k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Dec 14 2021 *)
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a(n) = sum(k=1, n, k*(n\k)^n);
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a(n) = sum(k=1, n, k*sumdiv(k, d, (d^n-(d-1)^n)/d));
A350107
a(n) = Sum_{k=1..n} k * floor(n/k)^2.
Original entry on oeis.org
1, 6, 14, 31, 45, 81, 101, 150, 191, 253, 285, 401, 439, 527, 623, 752, 802, 979, 1035, 1233, 1369, 1509, 1577, 1901, 2020, 2186, 2362, 2642, 2728, 3136, 3228, 3549, 3765, 3983, 4215, 4772, 4882, 5126, 5382, 5932, 6054, 6630, 6758, 7202, 7664, 7960, 8100, 8936
Offset: 1
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a[n_] := Sum[k * Floor[n/k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[2*k*DivisorSigma[0, k] - DivisorSigma[1, k], {k, 1, 100}]] (* Vaclav Kotesovec, Dec 16 2021 *)
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a(n) = sum(k=1, n, k*(n\k)^2);
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a(n) = sum(k=1, n, k*sumdiv(k, d, (2*d-1)/d));
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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)^2)/(1-x))
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a(n) = sum(k=1, n, 2*k*numdiv(k)-sigma(k));
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from math import isqrt
def A350107(n): return -(s:=isqrt(n))**3*(s+1)+sum((q:=n//k)*((k<<1)*((q<<1)+1)-q-1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 24 2023
A356124
Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).
Original entry on oeis.org
1, 1, 4, 1, 5, 8, 1, 7, 11, 15, 1, 11, 19, 23, 21, 1, 19, 41, 47, 33, 33, 1, 35, 103, 125, 77, 57, 41, 1, 67, 281, 395, 255, 149, 71, 56, 1, 131, 799, 1373, 1025, 555, 205, 103, 69, 1, 259, 2321, 5027, 4503, 2537, 905, 325, 130, 87, 1, 515, 6823, 18965, 20657, 12867, 4945, 1585, 442, 170, 99
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
4, 5, 7, 11, 19, 35, 67, ...
8, 11, 19, 41, 103, 281, 799, ...
15, 23, 47, 125, 395, 1373, 5027, ...
21, 33, 77, 255, 1025, 4503, 20657, ...
33, 57, 149, 555, 2537, 12867, 68969, ...
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T[n_, k_] := Sum[j^k * Binomial[Floor[n/j] + 1, 2], {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 28 2022 *)
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T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));
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T(n, k) = sum(j=1, n, j*sigma(j, k-1));
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from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A356124_T(n,k): return ((s:=isqrt(n))*(s+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)*(q+1))+(w*(bernoulli(k+1,q+1)-bernoulli(k+1))<<1) for w in range(1,s+1)))//(k+1)>>1
def A356124_gen(): # generator of terms
return (A356124_T(k+1,n-k-1) for n in count(1) for k in range(n))
A356124_list = list(islice(A356124_gen(),30)) # Chai Wah Wu, Oct 24 2023
A350108
a(n) = Sum_{k=1..n} k * floor(n/k)^3.
Original entry on oeis.org
1, 10, 32, 87, 153, 309, 443, 722, 1005, 1443, 1785, 2605, 3087, 3951, 4875, 6154, 6988, 8809, 9855, 12057, 13853, 16001, 17543, 21347, 23478, 26484, 29440, 33696, 36162, 41994, 44816, 50351, 54755, 59909, 64577, 73524, 77558, 84002, 90142, 100072, 105034
Offset: 1
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a[n_] := Sum[k * Floor[n/k]^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[(1 + 3*k)*DivisorSigma[1, k] - 3*k*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)
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a(n) = sum(k=1, n, k*(n\k)^3);
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a(n) = sum(k=1, n, k*sumdiv(k, d, (d^3-(d-1)^3)/d));
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my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^k)^2)/(1-x))
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from math import isqrt
def A350108(n): return -(s:=isqrt(n))**4*(s+1)+sum((q:=n//k)*(k**2*(3*(q+1))+k*(q*((q<<1)-3)-3)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 31 2023
Showing 1-4 of 4 results.