

A007897


a(n) is multiplicative with a(2) = 1; a(4) = 2; a(2^i) = 2^(i2)+2 if i>2; a(p^i) = 1+(p1)*p^(i1)/2 if prime p>2 and i>0.


4



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
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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], 20032018.


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^(e2)+2)), 1+(p1)*p^(e1)/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^(e2) + 2), 1 + p^(e1) * (p1) / 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: A164341 A124771 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 phifunction) to a(n).  N. J. A. Sloane, May 26 2014


STATUS

approved



