A067182 Smallest Fibonacci number with digit sum n, or -1 if no such number exists.

0, 1, 2, 3, 13, 5, -1, 34, 8, 144, 55

Offset: 0

Author

Amarnath Murthy, Jan 09 2002

Keywords

• sign
• base
• more

Comments

a(n) = Fibonacci(k) where k is the index of the first occurrence of n in A004090, or -1 if n never appears there. - N. J. A. Sloane, Dec 26 2016

Starting at n = 11, the terms a(11), a(12), ... are probably -1, -1, -1, 4181, -1, -1, 89, -1, 2584, 10946, 317811, 1597, 514229, 987, -1, -1, 46368, 28657, 196418, 2178309, 1346269, -1, 701408733, 3524578, 9227465, -1, 5702887, -1, -1, -1, 433494437, -1, 63245986, 39088169, -1, 267914296, -1, ... However, these -1's are only conjectural.

It appears that 0.9*n < A004090(n) < n for all but a few small n: In the range [0..10^5] the slope of A004090 is roughly 0.93. I conjecture that A004090(n) - n has 92 as its maximum, at n = 2619. This would prove that the given -1's are correct. - M. F. Hasler, Dec 26 2016

Joseph Myers and Don Reble proved that a(6) = -1 as follows (cf. Links): If the sum of digits of N is less than 9, then it equals the sum of digits of N modulo 10^k-1 for any k > 0. Now A000045 mod 9999 has period 600 (cf. A001175), and has no term equal to 6. - M. F. Hasler, Dec 28 2016

Examples

			a(14) = 4181, as it is the smallest Fibonacci number with a digit sum of 14.
		

Links

• Hans Havermann, Table of n, a(n) for n = 1..10000 (Note that this is not what would be called a b-file in the OEIS, since the -1 entries except for the first are conjectural, and a b-file may not contain conjectured values. - _N. J. A. Sloane_, Feb 05 2017)
• Hans Havermann, Table of n, a(n) for n = 1..10000 (the -1's, except for the first, are only conjectural). [Cached copy, with permission]
• Joseph Myers and Don Reble, Re: What are the possible digit-sums for Fibonacci numbers? (click "next" to see the second post), SeqFan list, Dec 27 2016

Crossrefs

Cf. A004090, A020995, A007953, A000045, A001175.

Programs

Mathematica

Take[#, 48] &@ Function[w, Function[t, {0}~Join~ReplacePart[t, Flatten@ Map[{#2 -> #1} & @@ # &, w]]]@ ConstantArray[0, w[[-1, -1]]]]@ Map[First, SplitBy[#, Last]] &@ SortBy[#, Last] &@ Table[{#, Total@ IntegerDigits@ #} &@ Fibonacci@ n, {n, 10^4}] (* Michael De Vlieger, Dec 28 2016 *)
a = 0; b = c = 1; t[] = -1; While[a < 10^1000, s = Plus @@ IntegerDigits[a]; If[s < 101 && t[s] == -1, t[s] = a]; a = b; b = c; c = a + b]; Array[t, 48, 0] (* _Robert G. Wilson v, Jan 25 2017 *)

PARI

A067182(n,a=1,b=-1)=-!for(k=0,n+99,sumdigits(a=b+b=a)==n&&return(a)) \\ M. F. Hasler, Dec 28 2016

Formula

a(n) = min { A000045(k) | A004090(k) = n } U { -1 }. - M. F. Hasler, Dec 26 2016

Extensions

More terms from Frank Ellermann, Jan 18 2002

More terms from Jason Earls, May 27 2002

Edited by M. F. Hasler, Dec 26 2016 and Dec 28 2016

Edited (including changing the value of a(n) for when no k exists from 0 to -1) by N. J. A. Sloane, Dec 29 2016 and Feb 05 2017