A020995 -id:A020995 - cp's OEIS frontend

A004090 Sum of digits of Fibonacci numbers.

0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 10, 17, 9, 8, 17, 7, 24, 22, 19, 14, 24, 20, 17, 28, 27, 19, 19, 29, 21, 23, 17, 31, 30, 34, 37, 35, 27, 35, 44, 43, 24, 31, 46, 41, 33, 29, 35, 37, 54, 55, 46, 29, 48, 41, 53, 58, 48, 52, 73, 44, 54, 53, 62, 61, 51, 67, 73, 59

Offset: 0

N. J. A. Sloane

a(n) and Fibonacci(n) are congruent modulo 9 which implies that (a(n) mod 9) is equal to (Fibonacci(n) mod 9) A007887(n). Thus (a(n) mod 9) is periodic with Pisano period A001175(9) = 24. - Hieronymus Fischer, Jun 25 2007

It appears that a(n) - n stays negative for n > 5832, which explains why A020995 is finite. - T. D. Noe, Mar 19 2012

T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Plot of a(n)-n for n = 0..100000

Cf. A000045 (Fibonacci), A007953 (digit sum), A030132 (digital root of A45), A010888 (digital root), A246558, A261587, A068500.

a004090 = a007953 . a000045 -- Reinhard Zumkeller, Nov 17 2014

[&+Intseq(Fibonacci(n)): n in [0..80] ]; // Vincenzo Librandi, Jun 18 2015

Table[Plus@@IntegerDigits@(Fibonacci[n]), {n, 0, 90}] (* Vincenzo Librandi, Jun 18 2015 *)

a(n)=sumdigits(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014

A004090 = lambda n: sum(int(d)for d in str(A000045(n))) # M. F. Hasler, Jul 07 2025

a(n) = Fibonacci(n) - 9*Sum_{k>0} floor(Fibonacci(n)/10^k). - Hieronymus Fischer, Jun 25 2007
a(n) = A007953(A000045(n)). - Reinhard Zumkeller, Nov 17 2014
A010888(a(n)) = A030132(n) == a(n) (mod 9). - M. F. Hasler, Jul 07 2025

A067515 Fibonacci numbers with index = digit sum.

Original entry on oeis.org

0, 1, 5, 55, 1346269, 9227465, 4052739537881, 498454011879264, 1672445759041379840132227567949787325, 18547707689471986212190138521399707760, 619220451666590135228675387863297874269396512

Offset: 1

Amarnath Murthy, Feb 14 2002

0 is a term as F(0) = 0.

Probably complete. See A020995 for indices, programs, etc. - T. D. Noe, Mar 04 2014

			55 is a term as F(10) = 55 and 5+5 = 10.

T. D. Noe, Table of n, a(n) for n = 1..20

Cf. A000045, A020995.

a(n) = A000045(A020995(n)). - Alois P. Heinz, Dec 29 2016

More terms from Sascha Kurz, Mar 18 2002

A020996 Numbers k such that the sum of the digits of Fibonacci(k) in base 12 is k.

Original entry on oeis.org

0, 1, 5, 13, 14, 89, 96, 123, 221, 387, 419, 550, 648, 749, 866, 892, 1105, 2037

Offset: 1

Sven Simon

No more terms < 100000. - Manfred Scheucher, Aug 03 2015

Manfred Scheucher, Sage Script
David Terr, On the Sums of Digits of Fibonacci Numbers, Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.

Cf. A020995 (base 10), A025490 (base 11).

filter:= proc(n) convert(convert(combinat:-fibonacci(n),base,12),`+`)=n end proc:
select(filter, [$1..3000]); # Robert Israel, Aug 03 2015

isok(n) = vecsum(digits(fibonacci(n), 12)) == n; \\ Michel Marcus, Feb 18 2015

a(1)=0 inserted by Sean A. Irvine, May 06 2019

A025490 Numbers k such that the sum of the digits of Fibonacci(k) in base 11 is k.

Original entry on oeis.org

0, 1, 5, 13, 41, 53, 55, 60, 61, 90, 97, 169, 185, 193, 215, 265, 269, 353, 355, 385, 397, 437, 481, 493, 617, 629, 630, 653, 713, 750, 769, 780, 889, 905, 960, 1013, 1025, 1045, 1205, 1320, 1405, 1435, 1501, 1620, 1650, 1657, 1705, 1735, 1769, 1793, 1913, 1981

Offset: 1

David W. Wilson

In his article, Terr estimated the number of terms to be 684 +- 26, which agrees with the found count of 710. - Sven Simon, Aug 06 2006

Sven Simon, Table of n, a(n) for n = 1..711 [May be complete]
David Terr, On the Sums of Digits of Fibonacci Numbers, Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.

Cf. A020995 (base 10), A020996 (base 12).

Cf. A025491 (with Lucas numbers).

[k:k in [0..2000]| &+Intseq(Fibonacci(k),11) eq k]; // Marius A. Burtea, Jun 09 2019

isok(n) = sumdigits(fibonacci(n), 11) == n; \\ Michel Marcus, Jun 08 2019

Title clarified and offset changed to 1 by Sean A. Irvine, May 06 2019

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

Original entry on oeis.org

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

Offset: 0

Amarnath Murthy, Jan 09 2002

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

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

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

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

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 *)

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

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

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

A383045 Integers k for which the sum of digits of Fibonacci(k) is a Fibonacci number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 13, 28, 33, 49, 85, 94, 107, 286, 299, 366, 421, 422, 443, 657, 2807, 4483, 4531, 18694, 49140, 79033, 79850, 80290, 128306, 129049, 129618, 208245, 338888, 546571, 882766, 883822, 886342

Offset: 1

Michel Marcus, Apr 14 2025

			Fibonacci(8) = 21 and sumdigits(21) = 3, a Fibonacci number, so 8 is a term.

Cf. A000045, A020995, A067515, A004090, A178837, A382053.

q:= n-> (t-> issqr(t+4) or issqr(t-4))(5*add(i, i=convert(combinat[fibonacci](n), base, 10))^2):
select(q, [$0..4600])[];  # Alois P. Heinz, Jul 15 2025

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; Select[Range[0, 1000], fibQ[DigitSum[Fibonacci[#]]] &] (* Amiram Eldar, Apr 14 2025 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(k) = isfib(sumdigits(fibonacci(k)));

a(36)-a(39) from Amiram Eldar, Apr 14 2025

cp's OEIS Frontend

A004090 Sum of digits of Fibonacci numbers.

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A067515 Fibonacci numbers with index = digit sum.

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A020996 Numbers k such that the sum of the digits of Fibonacci(k) in base 12 is k.

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A025490 Numbers k such that the sum of the digits of Fibonacci(k) in base 11 is k.

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

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A383045 Integers k for which the sum of digits of Fibonacci(k) is a Fibonacci number.

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