A308504
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).
Original entry on oeis.org
1, 1, 5, 1, 9, 28, 1, 17, 82, 273, 1, 33, 244, 1057, 3126, 1, 65, 730, 4161, 15626, 47450, 1, 129, 2188, 16513, 78126, 282252, 823544, 1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173
Offset: 1
a(4) = a(2*3/2 + 1) = sigma_3(1) = 1.
a(5) = a(2*3/2 + 2) = sigma_3(2) = 1^3 + 2^3 = 9.
a(6) = a(2*3/2 + 3) = sigma_3(3) = 1^3 + 3^3 = 28.
Square array begins:
1, 1, 1, 1, 1, ...
5, 9, 17, 33, 65, ...
28, 82, 244, 730, 2188, ...
273, 1057, 4161, 16513, 65793, ...
3126, 15626, 78126, 390626, 1953126, ...
47450, 282252, 1686434, 10097892, 60526250, ...
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T[n_, k_] := DivisorSum[n, #^(n+k) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)
A296601
L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^k) = Sum_{n>=1} a(n)*x^n/n.
Original entry on oeis.org
1, 9, 28, 81, 126, 330, 344, 833, 973, 1754, 1332, 5034, 2198, 5658, 8688, 13313, 4914, 28779, 6860, 54106, 45752, 33482, 12168, 254954, 93751, 78906, 255880, 505698, 24390, 1510700, 29792, 1671169, 1791312, 647114, 2819544, 12637371, 50654, 2282346, 14779520, 34058298, 68922, 68084220
Offset: 1
L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 81*x^4/4 + 126*x^5/5 + 330*x^6/6 + 344*x^7/7 + 833*x^8/8 + 973*x^9/9 + ...
exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 42*x^4 + 103*x^5 + 289*x^6 + 690*x^7 + 1771*x^8 + 4206*x^9 + ... + A266941(n)*x^n + ...
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nmax = 42; Rest[CoefficientList[Series[-Log[Product[(1 - k x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 42; Rest[CoefficientList[Series[Sum[k^3 x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[d^(n/d + 2), {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]
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N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^k)))) \\ Seiichi Manyama, Jun 02 2019
A308668
a(n) = Sum_{d|n} d^(n/d+n).
Original entry on oeis.org
1, 9, 82, 1089, 15626, 287010, 5764802, 135270401, 3487315843, 100244173394, 3138428376722, 107072686593858, 3937376385699290, 155601328490478978, 6568412173896940652, 295165920677390712833, 14063084452067724991010
Offset: 1
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a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)
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a(n) = sumdiv(n,d,d^(n/d+n));
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my(N=20, x='x+O('x^N)); Vec(x*deriv(-log(prod(k=1, N, (1-k*(k*x)^k)^(1/k)))))
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k^(k+1)*x^k))) \\ Seiichi Manyama, Mar 17 2021
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from sympy import divisors
def A308668(n): return sum(d**(n//d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022
Showing 1-3 of 3 results.