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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. Index entries for "core" sequences Index entries for sequences related to enumerating the rationals Index entries for sequences related to Stern's sequences 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 February 27 09:53 EST 2024. Contains 370377 sequences. (Running on oeis4.)