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A067182 Smallest Fibonacci number with digit sum n, or -1 if no such number exists. 1
0, 1, 2, 3, 13, 5, -1, 34, 8, 144, 55 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

Table of n, a(n) for n=0..10.

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

FORMULA

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A051298 A069870 A067180 * A342667 A191000 A085402

Adjacent sequences:  A067179 A067180 A067181 * A067183 A067184 A067185

KEYWORD

easy,sign,base

AUTHOR

Amarnath Murthy, Jan 09 2002

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

STATUS

approved

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Last modified April 18 15:04 EDT 2021. Contains 343089 sequences. (Running on oeis4.)