A004090 -id:A004090 - cp's OEIS frontend

A278833 Indices of records of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.

0, 6, 11, 16, 58, 178, 195, 273, 695, 862, 989, 2477, 2619

Offset: 1

M. F. Hasler, Dec 28 2016

For indices n = 1, 5, 83, 156 and 512, the value of the difference A004090(n) - n is the same as for the preceding record (see A278834). So the sequence of indices of records in the weak sense of >= would be (0, 1, 5, 6, 11, 16, 58, 83, 156, 178, 195, 273, 512, 695, 862, 989, 2477, 2619).

Conjectured to be finite and complete. Indeed it appears that 0.9*n < A004090(n) < n for all sufficiently large n.

Cf. A000045 (Fibonacci numbers), A004090 (their digital sums), A278834 (the record values corresponding to the indices listed here).

Except for the initial 0, this appears to be a subsequence of A264935.

Function[t, Flatten@ Map[First@ Position[t, #] - 1 &, Union@ Rest@ FoldList[Max, 0, t]]]@ Table[Plus @@ IntegerDigits@ Fibonacci@n - n, {n, 0, 10^4}] (* Michael De Vlieger, Dec 28 2016 *)

m=-1; for(k=0, 1e4, sumdigits(fibonacci(k))-k>m&&print1(k", ")+m=sumdigits(fibonacci(k))-k)

A278834 Record values of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.

Original entry on oeis.org

0, 2, 6, 8, 15, 30, 32, 40, 44, 46, 51, 57, 92

Offset: 1

M. F. Hasler, Dec 28 2016

For indices n = 1, 5, 83, 156 and 512, the value of the difference A004090(n) - n is the same as for the preceding record, namely (0, 0, 15, 15, 40), respectively. So the sequence of records in the weak sense of >= would be (0, 0, 0, 2, 6, 8, 15, 15, 15, 30, 32, 40, 40, 44, 46, 51, 57, 92).

Conjectured to be finite and complete. Indeed it appears that 0.9*n < A004090(n) < n for all sufficiently large n.

Cf. A000045 (Fibonacci numbers), A004090 (their digital sums), A278833 (the indices corresponding to the record values listed here), A264935.

Union@ Rest@ FoldList[Max, 0, #] &@ Table[Plus @@ IntegerDigits@ Fibonacci@ n - n, {n, 0, 10^4}] (* Michael De Vlieger, Dec 28 2016 *)

m=-1; for(k=0,1e4, sumdigits(fibonacci(k))-k>m && print1(m=sumdigits(fibonacci(k))-k,","))

A164947 Numbers n such that A004090(n)/A060384(n) is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 12, 16, 20, 21, 35, 56, 78, 84, 97, 125, 138, 184, 189, 300, 418, 437, 550

Offset: 1

Ctibor O. Zizka, Sep 01 2009

No further terms between 550 and 12000. [R. J. Mathar, Sep 27 2009]

No further terms up to 50000. - Michel Marcus, Apr 28 2018

Cf. A004090, A060384.

isok(n) = my(f=fibonacci(n), d=digits(f)); !(vecsum(d) % #d); \\ Michel Marcus, Apr 28 2018

14 more terms from R. J. Mathar, Sep 27 2009

A280185 a(n) = n - A004090(n), where A004090 is the sum of digits of the Fibonacci numbers A000045.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, -2, 3, 5, 2, 0, -6, 3, 5, -3, 8, -8, -5, -1, 5, -4, 1, 5, -5, -3, 6, 7, -2, 7, 6, 13, 0, 2, -1, -3, 0, 9, 2, -6, -4, 16, 10, -4, 2, 11, 16, 11, 10, -6, -6, 4, 22, 4, 12, 1, -3, 8, 5, -15, 15, 6, 8, 0, 2, 13, -2, -7, 8, 17, 4, 8, 25, 0, 9, -8, 10, 10, -9, -2, 21, -4, 2, 18, -15, 12, -4, 6, -10, 19, -5, 17, 23, 14, 28, 5, 4, 6, -3, 16, -2

Offset: 0

M. F. Hasler, Dec 28 2016

Conjectured to increase to infinity. It appears that the slope of A004090(n) is roughly 0.93, at least in the range 0..10^5.

I conjecture that this sequence takes its minimum at a(2619) = -92. - M. F. Hasler, Dec 30 2016

			a(2) = a(3) = 1 because A000045(2) = 1 = A004090(2), its digital sum, A000045(3) = 2 = A004090(3), and 2 - 1 = 3 - 2 = 1.

M. F. Hasler, Table of n, a(n) for n = 0..100000

Cf. A000045 (Fibonacci numbers), A004090 (their digital sums), A278834 & A278833 (record values of this sequence and corresponding indices), A264935.

Table[n - Total@ IntegerDigits@ Fibonacci@ n, {n, 0, 99}] (* Michael De Vlieger, Dec 28 2016 *)

A280185(n)=n-sumdigits(fibonacci(n))
/* To produce the b-file; can be used for searches or similar purpose, this is faster than to compute fib(n) anew for each term. */
b=-a=1;for(n=0,1e5,write("/tmp/A280185.txt",n" ",n-sumdigits(a=b+b=a)))

a(n) = n - A004090(n) = n - A007953(A000045(n)).

A030132 Digital root of Fibonacci(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8

Offset: 0

youngelder(AT)webtv.net (Ana)

Any initial pair (a(0), a(1)) of nonzero single-digit numbers enters a cycle of length 24, except for the 8 cases where 3 divides both a(0), a(1) and (a(0), a(1)) != (9, 9), which enter a cycle of length 8 and (9, 9), which is immediately periodic of period length 1. - Jonathan Vos Post, Dec 29 2005 [Corrected by Jianing Song, Apr 17 2021]
First term that differs from A004090 is a(10). In general, all terms of A004090 having one digit are the same in this sequence. - Alonso del Arte, Sep 16 2012
Decimal expansion of 12484270798876404618091 / 1111111111111111111111110 = 0.0[112358437189887641562819] (periodic). - Daniel Forgues, Feb 27 2017

			a(10) = 1 because F(10) = 55, and since 5 + 5 = 10 and 1 + 0 = 1 is the digital root of 55.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Robert Bruce Gray, Another context for the first 48 terms of this sequence, May 08 2025
S. Marivani and others, Digital Roots of Fibonacci Numbers: Problem 10974, Amer. Math. Monthly, 111 (No. 7, 2004), 628.
Colm Mulcahy, Gibonacci Bracelets.
Marc Renault, The Fibonacci sequence modulo m
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000045 (Fibonacci numbers), A010888 (digital roots), A004090, A007953, A030133.

Cf. A001175, A010077, A017641, A065076.

a030132 n = a030132_list !! n
a030132_list =
   0 : 1 : map a007953 (zipWith (+) a030132_list (tail a030132_list))
-- Reinhard Zumkeller, Aug 20 2011

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[If[n == 0, 0, digitalRoot[Fibonacci[n]]], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
Table[NestWhile[Total[IntegerDigits[#]]&, Fibonacci[n], # > 9 &], {n, 0, 90}] (* Harvey P. Dale, May 07 2012 *)
PadRight[{0},120,{9,1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1}] (* Harvey P. Dale, Jul 20 2024 *)

a(n)=if(n,(fibonacci(n)-1)%9+1,0) \\ Charles R Greathouse IV, Jan 23 2013

a(n + 1) = sum of digits of (a(n) + a(n - 1)).
Periodic with period 24 = A001175(9) given by {1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9}.
a(n) + a(n + 1) = A010077(n + 4); a(A017641(n)) = 9. - Reinhard Zumkeller, Jul 04 2007
G.f.: x*( -1 -x -2*x^2 -3*x^3 -5*x^4 -8*x^5 -4*x^6 -3*x^7 -7*x^8 -x^9 -8*x^10 -9*x^11 -8*x^12 -8*x^13 -7*x^14 -6*x^15 -4*x^16 -x^17 -5*x^18 -6*x^19 -2*x^20 -8*x^21 -x^22 -9*x^23 ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Feb 08 2013

Entry revised by N. J. A. Sloane, Aug 29 2004

A010077 a(n) = sum of digits of a(n-1) + sum of digits of a(n-2); a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10

Offset: 0

N. J. A. Sloane, Leonid Broukhis

The digital sum analog (in base 10) of the Fibonacci recurrence. - Hieronymus Fischer, Jun 27 2007
a(n) and Fibonacci(n) = A000045(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 the Pisano period A001175(9)=24. - Hieronymus Fischer, Jun 27 2007
a(n) == A004090(n) (mod 9) (A004090(n) = digital sum of Fibonacci(n)). - Hieronymus Fischer, Jun 27 2007
For general bases p > 2, we have the inequality 2 <= a(n) <= 2p-3 (for n > 2). Actually, a(n) <= 17 = A131319(10) for the base p=10. - Hieronymus Fischer, Jun 27 2007

David A. Corneth, Table of n, a(n) for n = 0..9999
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A007612, A065076.
Cf. A000045, A010073, A010074, A010075, A010076.
Cf. A131294, A131295, A131296, A131297, A131318, A131319, A131320.

a[0] = 0; a[1] = 1; a[n_] := a[n] = Apply[ Plus, IntegerDigits[ a[n - 1] ]] + Apply[ Plus, IntegerDigits[ a[n - 2] ]]; Table[ a[n], {n, 0, 100} ]
nxt[{a_,b_}]:={b, Total[IntegerDigits[a]]+Total[IntegerDigits[b]]}; NestList[ nxt,{0,1},80][[All,1]] (* Harvey P. Dale, Apr 15 2018 *)

first(n) = {n = max(n, 2); my(res = vector(n)); res[2] = 1; for(i = 3, n, res[i] = sumdigits(res[i-1]) + sumdigits(res[i-2]) ); res } \\ David A. Corneth, May 26 2021

Periodic from n=3 with period 24. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = A030132(n-4) + A030132(n-3) for n>3. - Reinhard Zumkeller, Jul 04 2007
a(n) = a(n-1) + a(n-2) - 9*(floor(a(n-1)/10) + floor(a(n-2)/10)). - Hieronymus Fischer, Jun 27 2007
a(n) = floor(a(n-1)/10) + floor(a(n-2)/10) + (a(n-1) mod 10) + (a(n-2) mod 10). - Hieronymus Fischer, Jun 27 2007
a(n) = A059995(a(n-1)) + A059995(a(n-2)) + A010879(a(n-1)) + A010879(a(n-2)). - Hieronymus Fischer, Jun 27 2007
a(n) = Fibonacci(n) - 9*Sum_{k=2..n-1} Fibonacci(n-k+1)*floor(a(k)/10) where Fibonacci(n) = A000045(n). - Hieronymus Fischer, Jun 27 2007

A131295 a(n)=ds_4(a(n-1))+ds_4(a(n-2)), a(0)=0, a(1)=1; where ds_4=digital sum base 4.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3

Offset: 0

Hieronymus Fischer, Jun 27 2007

The digital sum analog (in base 4) of the Fibonacci recurrence.
When starting from index n=3, periodic with Pisano period A001175(3)=8.
Also a(n)==A004090(n) modulo 3 (A004090(n)=digital sum of Fib(n)).
For general bases p>2, the inequality 2<=a(n)<=2p-3 holds for n>2. Actually, a(n)<=5=A131319(4) for the base p=4.
a(n) and Fib(n)=A000045(n) are congruent modulo 3 which implies that (a(n) mod 3) is equal to (Fib(n) mod 3)=A082115(n-1) (for n>0). Thus (a(n) mod 3) is periodic with the Pisano period = A001175(3)=8 too. - Hieronymus Fischer

			a(8)=3, since a(6)=5=11(base 4), ds_4(5)=2,
a(7)=4=10(base 4), ds_4(4)=1 and so a(8)=2+1.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for Colombian or self numbers and related sequences

Cf. A000045, A010073, A010074, A010075, A010076, A010077, A131294, A131296, A131297, A131318, A131319, A131320.

nxt[{a_,b_}]:={b,Total[IntegerDigits[a,4]]+Total[IntegerDigits[b,4]]}; NestList[ nxt,{0,1},110][[All,1]] (* Harvey P. Dale, Jul 30 2018 *)

a(n)=a(n-1)+a(n-2)-3*(floor(a(n-1)/4)+floor(a(n-2)/4)).
a(n)=floor(a(n-1)/4)+floor(a(n-2)/4)+(a(n-1)mod 4)+(a(n-2)mod 4).
a(n)=A002265(a(n-1))+A002265(a(n-2))+A010873(a(n-1))+A010873(a(n-2)).
a(n)=Fib(n)-3*sum{1A000045(n).

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

Original entry on oeis.org

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

Offset: 1

Sven Simon

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

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

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

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.

Cf. A000045, A067515, A004090.

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 *)

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

A246558 Product of the digits of the n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 3, 2, 12, 25, 72, 16, 18, 147, 0, 504, 315, 320, 32, 1260, 0, 49, 3360, 3456, 0, 162, 1728, 168, 720, 0, 7776, 0, 33600, 0, 30240, 0, 15680, 0, 311040, 0, 0, 326592, 435456, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102060, 3951360, 24883200, 1411200

Offset: 0

Indrani Das, Nov 12 2014

a(n) > 0 iff n in A076564.

Probably, the last nonzero term is a(184). - Giovanni Resta, Jul 14 2015

			Fibonacci(7) = 13, thus a(7) = 1*3 = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000045, A004090, A076564.
Cf. A007954, A259773.
Cf. A261598.

a246558 = a007954 . a000045 -- Reinhard Zumkeller, Nov 17 2014

[0] cat [&*Intseq(Fibonacci(n)): n in [1..100]]; // Vincenzo Librandi, Jan 04 2020

Array[Times@@IntegerDigits@Fibonacci[#]&, 100, 0] (* Vincenzo Librandi, Jan 04 2020 *)

a(n) = if (n, vecprod(digits(fibonacci(n))), 0); \\ Michel Marcus, Feb 11 2025

a(n) = A007954(A000045(n)). - Reinhard Zumkeller, Nov 17 2014

A139374 Digit sum of Lucas numbers.

Original entry on oeis.org

2, 1, 3, 4, 7, 2, 9, 11, 11, 13, 6, 19, 7, 8, 15, 14, 11, 16, 27, 25, 16, 23, 21, 26, 20, 28, 21, 22, 25, 29, 36, 20, 38, 40, 24, 28, 34, 26, 33, 23, 38, 34, 54, 43, 52, 41, 30, 62, 47, 46, 39, 49, 43, 47, 45, 47, 47, 58, 33, 73, 43, 53, 33, 68, 56, 70, 45, 43

Offset: 0

Parthasarathy Nambi, Jun 08 2008

			15127 is the 20th Lucas number (A000032(20)) with digit sum 16, so a(20)=16.

Irene Sermon, Table of n, a(n) for n = 0..10000

Cf. A000032, A004090, A007953.

[&+Intseq(Lucas(n)): n in [0..80]]; // Vincenzo Librandi, Jul 13 2015

Table[Total[IntegerDigits[LucasL[n]]], {n, 0, 100}] (* T. D. Noe, Oct 28 2013 *)

a(n) = A007953(A000032(n)). - Michel Marcus, Jul 13 2015

cp's OEIS Frontend

A278833 Indices of records of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.

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A278834 Record values of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.

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A164947 Numbers n such that A004090(n)/A060384(n) is an integer.

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A280185 a(n) = n - A004090(n), where A004090 is the sum of digits of the Fibonacci numbers A000045.

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A030132 Digital root of Fibonacci(n).

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A010077 a(n) = sum of digits of a(n-1) + sum of digits of a(n-2); a(0) = 0, a(1) = 1.

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A131295 a(n)=ds_4(a(n-1))+ds_4(a(n-2)), a(0)=0, a(1)=1; where ds_4=digital sum base 4.

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

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