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
A356239
a(n) = Sum_{k=1..n} k^n * sigma_0(k).
Original entry on oeis.org
1, 9, 71, 963, 9873, 231749, 2976863, 86348423, 1824883450, 55584932826, 1104642697680, 64932555347084, 1366828157222090, 61273696016238014, 2581786206601959958, 129797968403021602450, 3678372903755436314440, 295835829367866540495396
Offset: 1
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f:= proc(n) local k; add(k^n * numtheory:-tau(k),k=1..n) end proc:
map(f, [$1..30]); # Robert Israel, Jan 21 2024
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a[n_] := Sum[k^n * DivisorSigma[0, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)
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a(n) = sum(k=1, n, k^n*sigma(k, 0));
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a(n) = sum(k=1, n, k^n*sum(j=1, n\k, j^n));
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from math import isqrt
from sympy import bernoulli
def A356239(n): return (-(bernoulli(n+1, (s:=isqrt(n))+1)-(b:=bernoulli(n+1)))**2//(n+1) + sum(k**n*(bernoulli(n+1, n//k+1)-b)<<1 for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023
A356131
a(n) = Sum_{k=1..n} (k - 1)^n * binomial(floor(n/k)+1,2).
Original entry on oeis.org
0, 1, 9, 100, 1302, 20648, 377022, 7921039, 186926431, 4916562309, 142373072781, 4506381442625, 154721361953489, 5729251983077521, 227585590018322461, 9654855432715969784, 435659531345223039702, 20836069677785611552293
Offset: 1
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a[n_] := Sum[(k - 1)^n * Binomial[Floor[n/k]+1, 2], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)
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a(n) = sum(k=1, n, (k-1)^n*binomial((n\k)+1, 2));
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a(n) = sum(k=1, n, k*(sigma(k, n-1)-(n\k)^n));
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a(n) = sum(k=1, n, k*sumdiv(k, d, (d-1)^n/d));
A356243
a(n) = Sum_{k=1..n} k^2 * sigma_{n-2}(k).
Original entry on oeis.org
1, 9, 49, 447, 4607, 71009, 1210855, 24980627, 575624572, 14958422046, 427890493960, 13431874937840, 457651929853662, 16844143705998554, 665499756005678382, 28102799297908820326, 1262909308355648335240, 60183118566605371095996
Offset: 1
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a[n_] := Sum[k^2 * DivisorSigma[n - 2, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)
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a(n) = sum(k=1, n, k^2*sigma(k, n-2));
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a(n) = sum(k=1, n, k^n*sum(j=1, n\k, j^2));
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from math import isqrt
from sympy import bernoulli
def A356243(n): return (((s:=isqrt(n))*(s+1)*(2*s+1))*((b:=bernoulli(n+1))-bernoulli(n+1, s+1)) + sum(k**n*(n+1)*((q:=n//k)*(q+1)*(2*q+1))+6*k**2*(bernoulli(n+1,q+1)-b) for k in range(1,s+1)))//(n+1)//6 # Chai Wah Wu, Oct 21 2023
A356130
a(n) = Sum_{k=1..n} sigma_{n-1}(k).
Original entry on oeis.org
1, 4, 16, 111, 999, 12513, 185683, 3316418, 67810767, 1576561677, 40862702931, 1171104916405, 36722498575799, 1251419967587955, 46034784688102781, 1818440444592581068, 76763036794222996512, 3448830049286378614987, 164309958491233496689189
Offset: 1
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a[n_] := Sum[DivisorSigma[n-1, k], {k, 1, n}]; Array[a, 19] (* Amiram Eldar, Jul 28 2022 *)
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a(n) = sum(k=1, n, sigma(k, n-1));
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a(n) = sum(k=1, n, k^(n-1)*(n\k));
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from math import isqrt
from sympy import bernoulli
def A350130(n): return (((s:=isqrt(n))+1)*((b:=bernoulli(n))-bernoulli(n, s+1))+sum(k**(n-1)*n*((q:=n//k)+1)-b+bernoulli(n, q+1) for k in range(1,s+1)))//n if n>1 else 1 # Chai Wah Wu, Oct 21 2023
Showing 1-5 of 5 results.