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 A038567 Denominators in canonical bijection from positive integers to positive rationals <= 1. 39
 1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS n occurs phi(n) times (cf. A000010). Least k such that phi(1) + phi(2) + phi(3) + ... + phi(k) >= n. - Benoit Cloitre, Sep 17 2002 Sum of numerator and denominator of fractions arranged by Cantor's ordering (1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, 6/1, ...) with equivalent fractions removed. - Ron R. King, Mar 07 2009 [This applies to a(1, 2, ...) without initial term a(0) = 1 which could correspond to 0/1. - Editor's Note.] Care has to be taken in considering the offset which may be 0 or 1 in related sequences (see crossrefs), e.g., A038568 & A038569 also have offset 0, in A038566 offset has been changed to 1. - M. F. Hasler, Oct 18 2021 REFERENCES S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123. H. Lauwerier, Fractals, Princeton Univ. Press, p. 23. LINKS David Wasserman, Table of n, a(n) for n = 0..100000 Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633. FORMULA From Henry Bottomley, Dec 18 2000: (Start) a(n) = A020652(n) + A020653(n) for all n > 0, e.g., a(1) = 2 = 1 + 1 = A020652(1) + A020653(1). [Corrected and edited by M. F. Hasler, Dec 10 2021] n = a(A015614(n)) = a(A002088(n)) - 1 = a(A002088(n-1)). (End) a(n) = A002024(A169581(n)). - Reinhard Zumkeller, Dec 02 2009 a(A002088(n)) = n for n > 1. - Reinhard Zumkeller, Jul 29 2012 a(n) = A071912(2*n+1). - Reinhard Zumkeller, Dec 16 2013 EXAMPLE Arrange fractions by increasing denominator then by increasing numerator: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ...: this is A038566/A038567. MAPLE with (numtheory): A038567 := proc (n) local sum, k; sum := 1: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com) MATHEMATICA a[n_] := (k = 0; While[ Total[ EulerPhi[ Range[k]]] <= n, k++]; k); Table[ a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 08 2011, after Pari *) Flatten[Table[Table[n, {EulerPhi[n]}], {n, 20}]] (* Harvey P. Dale, Mar 12 2013 *) PROG (PARI) a(n)=if(n<0, 0, s=1; while(sum(i=1, s, eulerphi(i)) genericTake (a000010 x) \$ repeat x) [1..] -- Reinhard Zumkeller, Dec 16 2013, Jul 29 2012 (Python) from sympy import totient def a(n):     s=1     while sum(totient(i) for i in range(1, s + 1))

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Last modified November 26 08:23 EST 2022. Contains 358354 sequences. (Running on oeis4.)