A007897 a(n) is multiplicative with a(2) = 1; a(4) = 2; a(2^i) = 2^(i-2)+2 if i>2; a(p^i) = 1+(p-1)*p^(i-1)/2 if prime p>2 and i>0.

1, 1, 2, 2, 3, 2, 4, 4, 4, 3, 6, 4, 7, 4, 6, 6, 9, 4, 10, 6, 8, 6, 12, 8, 11, 7, 10, 8, 15, 6, 16, 10, 12, 9, 12, 8, 19, 10, 14, 12, 21, 8, 22, 12, 12, 12, 24, 12, 22, 11, 18, 14, 27, 10, 18, 16, 20, 15, 30, 12, 31, 16, 16, 18, 21, 12, 34, 18, 24, 12, 36, 16, 37, 19, 22, 20, 24, 14, 40, 18, 28

Offset

1,3

Comments

From Jeffrey Shallit, Jun 14 2018: (Start)

Except for first term, the same as A180783.

Equal to the number of elements x relatively prime to n such that x mod n >= x^(-1) mod n. (End)

References

Felix Weinstein, The Fibonacci Partitions, preprint, 1995.

Links

Table of n, a(n) for n=1..81.

F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2018.

Formula

Dirichlet g.f.: zeta(s) * zeta(s-1) * ((2 - 2^(s+2) + 2^(2*s+1) - 1/2^(2*s-2))/(2^(2*s+1) - 3*2^s - 1)) * Product_{p prime} (1 - (1/p^(s-1) + 1/p^s - 1/p^(2*s-1) + 1/p^(2*s))/2). - Amiram Eldar, Nov 09 2023

Example

G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 2*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + ...

Mathematica

a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ Apply[ Function[ {p, e}, If[p == 2, If[e < 3, e, 2^(e - 2) + 2], 1 + p^(e - 1) (p - 1)/2]], FactorInteger @ n, 1]]; (* Michael Somos, May 26 2014 *)

Prog

(PARI) ap(p, e) = if (p==2, if (e==1, 1, if (e==2, 2, 2^(e-2)+2)), 1+(p-1)*p^(e-1)/2);
a(n) = { my(f = factor(n)); prod(i=1, #f~, ap(f[i, 1], f[i, 2])); } \\ Michel Marcus, Apr 19 2014
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, if( e<3, e, 2^(e-2) + 2), 1 + p^(e-1) * (p-1) / 2))))}; /* Michael Somos, May 26 2014 */
(PARI) {a(n) = if( n<1, 0, direuler( p = 2, n, if( p>2, 1 / (1 - X) + (p - 1) / 2 * X / (1 - p*X), (1 + X^2) / (1 - X) + p * X^3 / (1 - p*X))) [n])}; /* Michael Somos, May 26 2014 */

Crossrefs

Cf. A007896, A007898, A180783.

Sequence in context: A124771 A334299 A066589 * A180783 A290731 A106289

Adjacent sequences: A007894 A007895 A007896 * A007898 A007899 A007900

Keyword

nonn,mult

Author

Felix Weinstein (wain(AT)ana.unibe.ch), Dec 11 1999

Extensions

Definition corrected by Michel Marcus, Apr 19 2014

Changed name from phi(n) (which caused much confusion with the Euler phi-function) to a(n). - N. J. A. Sloane, May 26 2014

Status

approved