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
A364269
a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.
Original entry on oeis.org
1, 41, 311, 1655, 4905, 15705, 32855, 76375, 142714, 272714, 435096, 797976, 1171466, 1857466, 2734966, 4131702, 5556472, 8210032, 10692990, 15060990, 19691490, 26186770, 32635280, 44385680, 54557555, 69497155, 85637215, 108686815, 129222353, 164322353
Offset: 1
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Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* Amiram Eldar, Oct 20 2023 *)
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f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));
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def A364269(n): return sum(k*(k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023
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from math import isqrt
def A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023
A356249
a(n) = Sum_{k=1..n} (k * floor(n/k))^3.
Original entry on oeis.org
1, 16, 62, 219, 405, 1053, 1523, 2948, 4407, 7041, 8703, 15283, 17949, 24657, 32685, 44806, 50536, 70687, 78573, 105411, 125879, 149879, 163565, 222425, 247476, 286134, 327634, 396258, 423084, 532236, 564818, 664763, 738095, 821693, 904937, 1107618, 1162268, 1277588, 1395760
Offset: 1
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a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jul 31 2022 *)
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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^3*sumdiv(k, d, 1-(1-1/d)^3));
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my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))
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from math import isqrt
def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023
A364268
a(n) = Sum_{k=1..n} k^2*sigma_2(k), where sigma_2 is A001157.
Original entry on oeis.org
1, 21, 111, 447, 1097, 2897, 5347, 10787, 18158, 31158, 45920, 76160, 104890, 153890, 212390, 299686, 383496, 530916, 661598, 879998, 1100498, 1395738, 1676108, 2165708, 2572583, 3147183, 3744963, 4568163, 5276285, 6446285, 7370767, 8768527, 10097107
Offset: 1
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Accumulate[Table[n^2*DivisorSigma[2, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)
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f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=2, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));
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def A364268(n): return sum(k**4*(m:=n//k)*(m+1)*((m<<1)+1)//6 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023
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from math import isqrt
def A364268(n): return (((s:=isqrt(n))*(s+1)*(2*s+1))**2*(1-3*s*(s+1))//6 + sum((q:=n//k)*(q+1)*(2*q+1)*k**2*(5*k**2+3*q*(q+1)-1) for k in range(1,s+1)))//30 # Chai Wah Wu, Oct 21 2023
A364194
a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.
Original entry on oeis.org
1, 25, 133, 581, 1331, 3923, 6667, 14347, 23824, 41824, 57796, 106180, 136938, 202794, 283794, 410770, 499204, 726652, 863832, 1199832, 1496184, 1879512, 2171520, 3000960, 3485335, 4223527, 5010847, 6240159, 6971829, 8915829, 9869141, 11933525, 13658501
Offset: 1
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Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)
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f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=1) = sum(k=1, n, k^(s+t)*f(n\k, s));
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def A364194(n): return sum((k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023
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from math import isqrt
def A364194(n): return ((((s:=isqrt(n))*(s + 1))**3*(2*s+1)*(1-3*s*(s+1))>>1) + sum((q:=n//k)*(q+1)*k**3*(q*(15*k+q*(15*k+12*q+18)+2)-2) for k in range(1,s+1)))//60 # Chai Wah Wu, Oct 21 2023
Showing 1-5 of 5 results.