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A343510
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Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.
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13
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1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 9, 1, 33, 83, 74, 29, 15, 1, 65, 245, 274, 129, 55, 13, 1, 129, 731, 1058, 629, 261, 55, 20, 1, 257, 2189, 4162, 3129, 1411, 349, 92, 21, 1, 513, 6563, 16514, 15629, 8085, 2407, 596, 105, 27, 1, 1025, 19685, 65794, 78129, 47515, 16813, 4388, 789, 145, 21
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. of column k: Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^(k+1).
T(n,k) = Sum_{d|n} phi(n/d) * d^k.
T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).
Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
T(n,k) = Sum_{j=1..n} (n/gcd(n,j))^k*phi(gcd(n,j))/phi(n/gcd(n,j)). - Richard L. Ollerton, May 10 2021
T(n,k) = Sum_{1 <= j_1, j_2, ..., j_k <= n} gcd(j_1, j_2, ..., j_k)^2 = Sum_{d divides n} d * J_k(n/d), where J_k(n) denotes the k-th Jordan totient function. - Peter Bala, Jan 29 2024
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EXAMPLE
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G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
3, 5, 9, 17, 33, 65, 129, ...
5, 11, 29, 83, 245, 731, 2189, ...
8, 22, 74, 274, 1058, 4162, 16514, ...
9, 29, 129, 629, 3129, 15629, 78129, ...
15, 55, 261, 1411, 8085, 47515, 282381, ...
13, 55, 349, 2407, 16813, 117655, 823549, ...
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MATHEMATICA
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T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * #^k &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)
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PROG
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(PARI) T(n, k) = sum(j=1, n, gcd(j, n)^k);
(PARI) T(n, k) = sumdiv(n, d, eulerphi(n/d)*d^k);
(PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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