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

%I A067182 #91 Apr 22 2025 09:08:36
%S A067182 0,1,2,3,13,5,-1,34,8,144,55
%N A067182 Smallest Fibonacci number with digit sum n, or -1 if no such number exists.
%C A067182 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
%C A067182 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.
%C A067182 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
%C A067182 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
%H A067182 Hans Havermann, <a href="http://chesswanks.com/seq/b067182.txt">Table of n, a(n) for n = 1..10000</a> (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)
%H A067182 Hans Havermann, <a href="/A067182/a067182.txt">Table of n, a(n) for n = 1..10000</a> (the -1's, except for the first, are only conjectural). [Cached copy, with permission]
%H A067182 Joseph Myers and Don Reble, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-December/017192.html">Re: What are the possible digit-sums for Fibonacci numbers?</a> (click "next" to see the second post), SeqFan list, Dec 27 2016
%F A067182 a(n) = min { A000045(k) | A004090(k) = n } U { -1 }. - _M. F. Hasler_, Dec 26 2016
%e A067182 a(14) = 4181, as it is the smallest Fibonacci number with a digit sum of 14.
%t A067182 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 *)
%t A067182 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 *)
%o A067182 (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
%Y A067182 Cf. A004090, A020995, A007953, A000045, A001175.
%K A067182 sign,base,more
%O A067182 0,3
%A A067182 _Amarnath Murthy_, Jan 09 2002
%E A067182 More terms from _Frank Ellermann_, Jan 18 2002
%E A067182 More terms from _Jason Earls_, May 27 2002
%E A067182 Edited by _M. F. Hasler_, Dec 26 2016 and Dec 28 2016
%E A067182 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