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 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. 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 (list; graph; refs; listen; history; text; internal format)
 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 F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2018. 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), 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

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Last modified May 9 00:09 EDT 2021. Contains 343685 sequences. (Running on oeis4.)