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A020995
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Numbers k such that the sum of the digits of Fibonacci(k) is k.
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6
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0, 1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222
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OFFSET
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1,3
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COMMENTS
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Since the number of digits in the k-th Fibonacci number ~ k*log_10(Golden Ratio), theoretically this sequence is infinite, but then the average density of those digits = ~ 0.208987. - Robert G. Wilson v
Robert Dawson of Saint Mary's University says it is likely that 2222 is the last term, as (assuming that the digits are equally distributed) the expected digit sum is ~ 0.9*k. - Stefan Steinerberger, Mar 12 2006 [Assuming that the average digit is (0+1+2+...+9)/10 = 9/2, the expected digit sum is ~ (9/2)*log_10((1+sqrt(5))/2)*k = 0.94044438...*k. - Jon E. Schoenfield, Aug 28 2022]
Bankoff's short paper lists the first seven terms. - T. D. Noe, Mar 19 2012
No more terms < 150000. - Manfred Scheucher, Aug 03 2015
If it exists, a(21) > 10^6. - Robert Price, May 26 2019
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REFERENCES
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Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 209.
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LINKS
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Table of n, a(n) for n=1..20.
Leon Bankoff, A Fibonacci Curiosity, Fibonacci Quarterly 14, Feb. 1976, p. 17.
Pat Ballew, Fibonacci Digit Sums, Pat's Blog, Sunday, 5 August 2012.
Ron Knott, The Mathematical Magic of the Fibonacci Numbers: Digit Sums
Manfred Scheucher, Sage Script
David Terr, On the Sums of Digits of Fibonacci Numbers, Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.
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EXAMPLE
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Fibonacci(10) = 55 and 5+5 = 10.
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MATHEMATICA
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Do[ If[ Apply[ Plus, IntegerDigits[ Fibonacci[n]]] == n, Print[n]], {n, 1, 10^5} ] (* Sven Simon *)
Do[ If[ Mod[ Fibonacci[n], 9] == Mod[n, 9], If[ Plus @@ IntegerDigits[ Fibonacci[n]] == n, Print[n]]], {n, 0, 10^6}] (* Robert G. Wilson v *)
Select[Range[0, 10^5], Plus @@ IntegerDigits[Fibonacci[ # ]] == # &] (* Ron Knott, Oct 30 2010 *)
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PROG
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(PARI) isok(n) = sumdigits(fibonacci(n)) == n; \\ Michel Marcus, Feb 18 2015
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CROSSREFS
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Cf. A000045, A067515, A004090.
Sequence in context: A056422 A032296 A052648 * A174467 A005201 A221304
Adjacent sequences: A020992 A020993 A020994 * A020996 A020997 A020998
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KEYWORD
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nonn,base,more
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AUTHOR
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Sven Simon
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STATUS
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approved
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