A227507 Table of p(a,n) read by antidiagonals, where p(a,n) = Sum_{k=1..n} gcd(k,n) exp(2 Pi i k a / n) is the Fourier transform of the greatest common divisor.

1, 3, 1, 5, 1, 1, 8, 2, 3, 1, 9, 2, 2, 1, 1, 15, 4, 4, 5, 3, 1, 13, 2, 4, 2, 2, 1, 1, 20, 6, 6, 4, 8, 2, 3, 1, 21, 4, 6, 5, 4, 2, 5, 1, 1, 27, 6, 8, 6, 6, 9, 4, 2, 3, 1, 21, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1, 40, 10, 12, 12, 12, 6, 15, 4, 8, 5, 3, 1, 25, 4, 10, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1, 39, 12, 8, 10, 12, 6

Offset

1,2

Comments

p(a,n) gives the number of pairs (i,j) of congruence classes modulo n, such that i*j = a mod n.

p(a,n) is a multiplicative function of n.

Links

Table of n, a(n) for n=1..97.

U. Abel, W. Awan, and V. Kushnirevych, A Generalization of the Gcd-Sum Function, J. Int. Seq. 16 (2013), #13.6.7.

Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, INTEGERS 13 (2013), A24.

Formula

The function can be written as a generalized Ramanujan sum: p(a,n) = Sum_{d|gcd(a,n)} d phi(n/d), where phi(n) denotes the totient function.

The rows of its table are equal to two of the diagonals: p(a,n) = p(n-a,n) = p(n+a,n).

p(0,n) = A018804(n), p(1,n) = A000010(n).

f(n) = Sum_{k=1..n} p(r,k)/k = Sum_{k=1..n} c_k(r)/k * floor(n/k), where c_k(r) denotes Ramanujan's sum (A054533(r)).

Example

1, 3, 5, 8, 9, 15, 13, 20, 21, 27
1, 1, 2, 2, 4, 2, 6, 4, 6, 4
1, 3, 2, 4, 4, 6, 6, 8, 6, 12
1, 1, 5, 2, 4, 5, 6, 4, 12, 4
1, 3, 2, 8, 4, 6, 6, 12, 6, 12
1, 1, 2, 2, 9, 2, 6, 4, 6, 9
The array G_d(n) of Abel et al. (with A018804 on the diagonal) starts as follows:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ,...
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3,...
2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2,...
2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 8, 2, 4, 2, 8,...
4, 4, 4, 4, 9, 4, 4, 4, 4, 9, 4, 4, 4, 4, 9, 4, 4, 4, 4, 9,...
2, 6, 5, 6, 2,15, 2, 6, 5, 6, 2,15, 2, 6, 5, 6, 2,15, 2, 6,...
6, 6, 6, 6, 6, 6,13, 6, 6, 6, 6, 6, 6,13, 6, 6, 6, 6, 6, 6,...
4, 8, 4,12, 4, 8, 4,20, 4, 8, 4,12, 4, 8, 4,20, 4, 8, 4,12,..
6, 6,12, 6, 6,12, 6, 6,21, 6, 6,12, 6, 6,12, 6, 6,21, 6, 6,...
4,12, 4,12, 9,12, 4,12, 4,27, 4,12, 4,12, 9,12, 4,12, 4,27,...
10,10,10,10,10,10,10,10,10,10,21,10,10,10,10,10,10,10,10,10,...
4, 8,10,16, 4,20, 4,16,10, 8, 4,40, 4, 8,10,16, 4,20, 4,16,...
12,12,12,12,12,12,12,12,12,12,12,12,25,12,12,12,12,12,12,12,...
... - R. J. Mathar, Jan 21 2018

Maple

p:=(a, n)->add(d*phi(n/d), d in divisors(gcd(a, n))):
seq(seq(p(a, n-a), a=0..n-1), n=1..10);

Crossrefs

Cf. A000010, A018804, A054532, A054533, A054534, A054535.

Sequence in context: A171232 A093423 A326454 * A134700 A085407 A378132

Adjacent sequences: A227504 A227505 A227506 * A227508 A227509 A227510

Keyword

nonn,mult,tabl

Author

Peter H van der Kamp, Jul 13 2013

Status

approved