A382995
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = Sum_{d|n} phi(n/d) * (-k)^(d-1).
Original entry on oeis.org
1, 1, 0, 1, -1, 3, 1, -2, 6, 0, 1, -3, 11, -8, 5, 1, -4, 18, -28, 20, 0, 1, -5, 27, -66, 85, -30, 7, 1, -6, 38, -128, 260, -238, 70, 0, 1, -7, 51, -220, 629, -1014, 735, -136, 9, 1, -8, 66, -348, 1300, -3108, 4102, -2216, 270, 0, 1, -9, 83, -518, 2405, -7750, 15631, -16452, 6585, -500, 11
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, -6, ...
3, 6, 11, 18, 27, 38, 51, ...
0, -8, -28, -66, -128, -220, -348, ...
5, 20, 85, 260, 629, 1300, 2405, ...
0, -30, -238, -1014, -3108, -7750, -16770, ...
7, 70, 735, 4102, 15631, 46662, 117655, ...
A343490
a(n) = Sum_{k=1..n} 4^(gcd(k, n) - 1).
Original entry on oeis.org
1, 5, 18, 70, 260, 1050, 4102, 16460, 65574, 262420, 1048586, 4195500, 16777228, 67112990, 268436040, 1073758360, 4294967312, 17179936830, 68719476754, 274878169880, 1099511636076, 4398047559730, 17592186044438, 70368748407000, 281474976711700, 1125899923619900
Offset: 1
-
N:= 30: # for a(1)..a(N)
G:= add(numtheory:-phi(k)*x^k/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Sep 11 2023
-
a[n_] := Sum[4^(GCD[k, n] - 1), {k, 1, n}]; Array[a, 26] (* Amiram Eldar, Apr 17 2021 *)
-
a(n) = sum(k=1, n, 4^(gcd(k, n)-1));
-
a(n) = sumdiv(n, d, eulerphi(n/d)*4^(d-1));
-
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-4*x^k)))
A343492
a(n) = Sum_{k=1..n} 5^(gcd(k, n) - 1).
Original entry on oeis.org
1, 6, 27, 132, 629, 3162, 15631, 78264, 390681, 1953774, 9765635, 48831564, 244140637, 1220718786, 6103516983, 30517656528, 152587890641, 762939850086, 3814697265643, 19073488283028, 95367431672037, 476837167968810, 2384185791015647, 11920929004069128
Offset: 1
-
a[n_] := Sum[5^(GCD[k, n] - 1), {k, 1, n}]; Array[a, 24] (* Amiram Eldar, Apr 17 2021 *)
-
a(n) = sum(k=1, n, 5^(gcd(k, n)-1));
-
a(n) = sumdiv(n, d, eulerphi(n/d)*5^(d-1));
-
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-5*x^k)))
Showing 1-3 of 3 results.