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 A054897 a(n) = Sum_{k>0} floor(n/8^k). 4
 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,17 COMMENTS Different from the highest power of 8 dividing n!, A090617. LINKS Hieronymus Fischer, Table of n, a(n) for n = 0..10000 FORMULA a(n) = floor(n/8) + floor(n/64) + floor(n/512) + floor(n/4096) + .... a(n) = (n - A053829(n))/7. From Hieronymus Fischer, Aug 14 2007: (Start) Recurrence: a(n) = floor(n/8) + a(floor(n/8)); a(8*n) = n + a(n); a(n*8^m) = n*(8^m-1)/7 + a(n). a(k*8^m) = k*(8^m-1)/7, for 0 <= k < 8, m >= 0. Asymptotic behavior: a(n) = n/7 + O(log(n)), a(n+1) - a(n) = O(log(n)); this follows from the inequalities below. a(n) <= (n-1)/7; equality holds for powers of 8. a(n) >= (n-7)/7 - floor(log_8(n)); equality holds for n=8^m-1, m>0. lim inf (n/7 - a(n)) = 1/7, for n -> oo. lim sup (n/7 - log_8(n) - a(n)) = 0, for n -> oo. lim sup (a(n+1) - a(n) - log_8(n)) = 0, for n -> oo. G.f.: g(x) = ( Sum_{k>0} x^(8^k)/(1-x^(8^k)) )/(1-x). (End) Partial sums of A244413. - R. J. Mathar, Jul 08 2021 EXAMPLE a(100) = 13. a(10^3) = 141. a(10^4) = 1427. a(10^5) = 14284. a(10^6) = 142855. a(10^7) = 1428569. a(10^8) = 14285710. a(10^9) = 142857138. MATHEMATICA Table[t=0; p=8; While[s=Floor[n/p]; t=t+s; s>0, p *= 8]; t, {n, 0, 100}] PROG (Python) def A054897(n): return (n-sum(int(d) for d in oct(n)[2:]))//7 # Chai Wah Wu, Jul 09 2022 (Magma) m:=8; function a(n) // a = A054897 if n eq 0 then return n; else return a(Floor(n/m)) + Floor(n/m); end if; end function; [a(n): n in [0..103]]; // G. C. Greubel, Apr 28 2023 (SageMath) m=8 # a = A054897 def a(n): return 0 if (n==0) else a(n//m) + (n//m) [a(n) for n in range(104)] # G. C. Greubel, Apr 28 2023 CROSSREFS Cf. A011371 and A054861 for analogs involving powers of 2 and 3. Cf. A054899, A067080, A098844, A132032. Sequence in context: A132292 A110656 A104407 * A261226 A003108 A279223 Adjacent sequences: A054894 A054895 A054896 * A054898 A054899 A054900 KEYWORD nonn AUTHOR Henry Bottomley, May 23 2000 EXTENSIONS Examples added by Hieronymus Fischer, Jun 06 2012 STATUS approved

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Last modified May 26 08:37 EDT 2024. Contains 372815 sequences. (Running on oeis4.)