A343516
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1 <= x_2 <= ... <= x_k <= n} gcd(x_1, x_2, ... , x_k, n).
Original entry on oeis.org
1, 1, 3, 1, 4, 5, 1, 5, 8, 8, 1, 6, 12, 15, 9, 1, 7, 17, 26, 19, 15, 1, 8, 23, 42, 39, 35, 13, 1, 9, 30, 64, 74, 76, 34, 20, 1, 10, 38, 93, 130, 153, 90, 56, 21, 1, 11, 47, 130, 214, 287, 216, 152, 63, 27, 1, 12, 57, 176, 334, 506, 468, 379, 191, 86, 21
Offset: 1
T(4,2) = gcd(1,1,4) + gcd(1,2,4) + gcd(2,2,4) + gcd(1,3,4) + gcd(2,3,4) + gcd(3,3,4) + gcd(1,4,4) + gcd(2,4,4) + gcd(3,4,4) + gcd(4,4,4) = 1 + 1 + 2 + 1 + 1 + 1 + 1 + 2 + 1 + 4 = 15.
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
3, 4, 5, 6, 7, 8, 9, ...
5, 8, 12, 17, 23, 30, 38, ...
8, 15, 26, 42, 64, 93, 130, ...
9, 19, 39, 74, 130, 214, 334, ...
15, 35, 76, 153, 287, 506, 846, ...
13, 34, 90, 216, 468, 930, 1722, ...
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T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * Binomial[k + # - 1, k] &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)
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T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));
A343553
a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).
Original entry on oeis.org
1, 3, 8, 26, 74, 287, 930, 3572, 12966, 49379, 184766, 710712, 2704168, 10427822, 40123208, 155289768, 601080406, 2334740919, 9075135318, 35352194658, 137846990678, 538302226835, 2104098963742, 8233721100024, 32247603765020, 126412458921072, 495918569262798
Offset: 1
a(3) = gcd(1,1,3) + gcd(1,2,3) + gcd(1,3,3) + gcd(2,2,3) + gcd(2,3,3) + gcd(3,3,3) = 1 + 1 + 1 + 1 + 1 + 3 = 8.
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a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + n - 2, n-1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
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a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-2, n-1));
A343565
a(n) = |{(x_1, x_2, ... , x_n) : 1 <= x_1 <= x_2 <= ... <= x_n <= n, gcd(x_1, x_2, ... , x_n, n) = 1}|.
Original entry on oeis.org
1, 2, 9, 30, 125, 428, 1715, 6270, 24255, 91367, 352715, 1345448, 5200299, 20019526, 77554749, 300295038, 1166803109, 4535971916, 17672631899, 68913247655, 269128640958, 1051984969598, 4116715363799, 16123381989000, 63205303195125, 247956558998878, 973469689288236
Offset: 1
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a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[# + n - 1, n] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
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a(n) = sumdiv(n, d, moebius(n/d)*binomial(d+n-1, n));
A345230
a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ..., x_n).
Original entry on oeis.org
0, 1, 4, 13, 44, 140, 512, 1782, 6652, 24682, 93599, 354341, 1359470, 5210328, 20098886, 77621774, 300797854, 1167164438, 4539201401, 17674941735, 68933414989, 269143872226, 1052114789548, 4116808923486, 16124224585644, 63205911146740, 247961982954952
Offset: 0
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a:= n-> coeff(series((1/(1-x))* add(numtheory[phi](k)
*x^k/(1-x^k)^n, k=1..n), x, n+1), x, n):
seq(a(n), n=0..26); # Alois P. Heinz, Jun 11 2021
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a[n_] := Sum[DivisorSum[k, EulerPhi[k/#] * Binomial[n + # - 2, n - 1] &], {k, 1, n}]; Array[a, 30, 0] (* Amiram Eldar, Jun 11 2021 *)
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a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(k/d)*binomial(d+n-2, n-1)));
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a(n) = sum(k=1, n, eulerphi(k)*binomial(n\k+n-1, n)); \\ Seiichi Manyama, Sep 13 2024
A338655
a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-1, d).
Original entry on oeis.org
1, 3, 5, 9, 9, 22, 13, 32, 35, 53, 21, 121, 25, 96, 177, 166, 33, 297, 37, 491, 417, 218, 45, 1002, 549, 297, 705, 1375, 57, 2418, 61, 1640, 1405, 491, 3887, 4659, 73, 606, 2233, 8156, 81, 8989, 85, 6189, 11955, 872, 93, 16550, 10387, 12927, 4757, 11111, 105, 22392, 25757
Offset: 1
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a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
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a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-1, d));
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my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^(k+1)))
Showing 1-5 of 5 results.