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, ...
-
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 *)
-
T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));
Original entry on oeis.org
1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, 63, 13, 56, 99, 89, 17, 154, 19, 269, 237, 132, 23, 509, 301, 182, 379, 783, 29, 1230, 31, 881, 813, 306, 2125, 2431, 37, 380, 1299, 4157, 41, 4822, 43, 3695, 6175, 552, 47, 8529, 5587, 6266, 2787
Offset: 1
a(4) = 5 = (1, 2, 0, 1) dot (1, 1, 2, 2) = (1 + 2 + 0 + 2), where row 4 of A156348 = (1, 2, 0, 1) and (1, 1, 2, 2) = the first 4 terms of Euler's phi function.
-
A156834 := proc(n)
add(A156348(n,k)*numtheory[phi](k),k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013
-
a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 2, #-1] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
-
a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-2, d-1)); \\ Seiichi Manyama, Apr 22 2021
-
my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*(x/(1-x^k))^k)) \\ Seiichi Manyama, Apr 22 2021
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
-
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
-
a[n_] := Sum[DivisorSum[k, EulerPhi[k/#] * Binomial[n + # - 2, n - 1] &], {k, 1, n}]; Array[a, 30, 0] (* Amiram Eldar, Jun 11 2021 *)
-
a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(k/d)*binomial(d+n-2, n-1)));
-
a(n) = sum(k=1, n, eulerphi(k)*binomial(n\k+n-1, n)); \\ Seiichi Manyama, Sep 13 2024
Showing 1-3 of 3 results.
Comments