A020995 Numbers k such that the sum of the digits of Fibonacci(k) is k.

0, 1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222

Offset: 1

Author

Sven Simon

Keywords

• nonn
• base
• more

Comments

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

Examples

			Fibonacci(10) = 55 and 5+5 = 10.
		

References

• Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 209.

Links

• 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.

Crossrefs

Cf. A000045, A067515, A004090.

Programs

Mathematica

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 *)

PARI

isok(n) = sumdigits(fibonacci(n)) == n; \\ Michel Marcus, Feb 18 2015