A004090 -id:A004090 - cp's OEIS frontend

A261587 Sum of sexagesimal digits of Fibonacci numbers in base-60 representation.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 30, 26, 56, 23, 20, 43, 63, 47, 51, 98, 31, 70, 101, 112, 95, 89, 125, 96, 103, 81, 125, 29, 95, 65, 101, 48, 149, 138, 169, 130, 122, 134, 138, 154, 174, 151, 148, 122, 152, 156, 131, 169, 241, 233, 179, 235, 178, 236

Offset: 0

Reinhard Zumkeller, Sep 09 2015

a(n) is the sum of the n-th row of table A261575.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Sexagesimal

Cf. A261575, A000045, A004090, A261598.

Haskell
```
a261587 = sum . a261575_row
```

a:= n-> add(i, i=convert((<<0|1>, <1|1>>^n)[1, 2], base, 60)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 22 2022

Table[Total[IntegerDigits[n,60]],{n,Fibonacci[Range[0,60]]}] (* Harvey P. Dale, Aug 02 2019 *)

a(n) = sumdigits(fibonacci(n), 60); \\ Michel Marcus, Jan 22 2022

A337448 The numbers k for which Fibonacci(k) are Niven numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 12, 18, 36, 54, 72, 84, 112, 120, 144, 160, 180, 198, 200, 216, 240, 243, 264, 286, 288, 299, 324, 358, 360, 468, 504, 528, 540, 576, 648, 720, 780, 816, 1008, 1020, 1044, 1200, 1248, 1260, 1500, 1602, 1824, 1872, 1917, 2160, 2184, 2760

Offset: 1

Marius A. Burtea, Sep 14 2020

For a(7) = 8, Fibonacci(8) = 21 and 21/digsum(21) = 7 is a prime number, so Fibonacci(8) is a Moran number (A001101). It seems that this is the only Moran number among the first 100000 Fibonacci numbers.

			Fibonacci(1) = 1 = A005349(1), so 1 is a term.
Fibonacci(8) = 21 = A005349(14), so 8 is a term.
Fibonacci(12) = 144 = A005349(8), so 12 is a term.
Fibonacci(18) = 2584 = A005349(514), so 18 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A000045, A004090, A001101, A005349, A117774, A337449.

niven:=func; [k:k in [1..2760]| niven(Fibonacci(k))];

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; Select[Range[3000], nivenQ[Fibonacci[#]] &] (* Amiram Eldar, Sep 15 2020 *)

isok(k) = my(f=fibonacci(k)); (f % sumdigits(f)) == 0; \\ Michel Marcus, Sep 15 2020

A112661 Sum of digits of sum of previous 3 terms.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4, 3, 2, 9, 5, 7, 3, 6, 7, 7, 2, 7, 7, 7, 3, 8, 9, 2, 10, 3, 6, 10, 10, 8, 10, 10, 10, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4, 3, 2, 9, 5, 7, 3, 6, 7, 7, 2, 7, 7, 7, 3, 8, 9, 2, 10, 3, 6, 10, 10, 8, 10, 10, 10, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4

Offset: 0

Jonathan Vos Post and Andrew Carmichael Post (andrewpost(AT)gmail.com), Dec 29 2005

Sum of digits, not iterated (i.e., not digital sum, reducing to a single digit) as we twice get a term of 10 which we do not reduce to 1. This is to tribonacci (A000073) as A030132 is to Fibonacci (A000045). This sequence has a preamble of 3 terms (1, 1, 1), then enters a cycle of length 39 (ending with 10, 10, 10).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000073, A004090, A007953, A010888, A030132.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Total@ IntegerDigits[a[n-1] + a[n-2] + a[n-3]]; a /@ Range[0, 93] (* Giovanni Resta, Jun 17 2016 *)

a(n+2) = sum of digits of (a(n) + a(n-1) + a(n-2)). a(n+2) = A007953(a(n) + a(n-1) + a(n-2)).

Data and name corrected by Giovanni Resta, Jun 17 2016

A068500 Sequence of Fibonacci numbers whose sum of decimal digits sets a new record.

Original entry on oeis.org

1, 2, 3, 5, 8, 55, 89, 987, 28657, 196418, 1346269, 3524578, 5702887, 39088169, 267914296, 4807526976, 7778742049, 139583862445, 591286729879, 1304969544928657, 5527939700884757, 99194853094755497, 83621143489848422977, 218922995834555169026, 927372692193078999176

Offset: 1

Shyam Sunder Gupta, Mar 25 2002

			a(8)=987 and 9+8+7=24 and sum of digits of any Fibonacci numbers < 987 is also less than 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..225

Cf. A000045, A007953, A004090.

a068500 n = a068500_list !! (n-1)
a068500_list = h 0 a004090_list a000045_list where
   h r (q:qs) (f:fs) = if q <= r then h r qs fs else f : h q qs fs
-- Reinhard Zumkeller, Oct 26 2015

terms = 30; Reap[For[n = k = 1; record = 0, n <= terms, k++, an = Fibonacci[k]; t = Total[IntegerDigits[an]]; If[t > record, record = t; Print["a(", n, ") = ", an]; Sow[an]; n++]]][[2, 1]] (* Jean-François Alcover, Apr 02 2017 *)
DeleteDuplicates[Table[{f,Total[IntegerDigits[f]]},{f,Fibonacci[Range[150]]}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 22 2024 *)

lista(nn) = {s = 0; for (n=1, nn, if ((ns = sumdigits(f=fibonacci(n))) > s, print1(f, ", "); s = ns););} \\ Michel Marcus, Apr 02 2017

A139563 Fibonacci numbers whose digit sum is a Lucas number.

Original entry on oeis.org

1, 2, 3, 13, 21, 34, 610, 196418, 1134903170, 20365011074, 15635695580168194910579363790217849593217, 1049252690665646467530632231274619718410203796555123147644873726135009824265250

Offset: 1

Parthasarathy Nambi, Jun 11 2008

Depending on whether the Lucas numbers are defined by A000032 or by A000204, one obtains this sequence here or A117766. - R. J. Mathar, Nov 03 2008

The next term (a(13)) has 108 digits. - Harvey P. Dale, Jul 01 2022

			196418 is a Fibonacci number whose digit sum 29 is a Lucas number.

Cf. A000032, A000045, A004090, A117766.

luQ[n_] := n==2 || Block[{i=1}, While[LucasL[i] < n, i++]; LucasL[i] == n]; Select[
Fibonacci[ Range[2, 400]], luQ[ Plus @@ IntegerDigits[#]] &] (* Giovanni Resta, Mar 15 2020 *)
Module[{nn=500,ln=LucasL[Range[0,20]]},Select[Fibonacci[Range[nn]],MemberQ[ ln,Total[ IntegerDigits[ #]]]&]]//Union (* Harvey P. Dale, Jul 01 2022 *)

1 added in front by R. J. Mathar, Nov 03 2008

More terms from Jinyuan Wang, Mar 15 2020

A216699 Digital root of cubes of Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8

Offset: 0

Ravi Bhandari, Sep 15 2012

This sequence repeats after every 8 terms, hence this is periodic with period 8.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A004090, A056570, A216676.

Table[NestWhile[Total[IntegerDigits[#]] &, Fibonacci[n]^3, # > 9 &], {n, 0, 86}] (* T. D. Noe, Oct 15 2012 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 8, 9, 8, 8, 1, 9},86]] (* Ray Chandler, Aug 25 2015 *)

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

A110390 a(n) = F(n) mod s(n) where s(n) is the sum of the digits of the n-th Fibonacci number F(n).

Original entry on oeis.org

1, 0, 6, 5, 4, 0, 1, 3, 1, 3, 13, 0, 9, 21, 6, 14, 13, 9, 13, 2, 1, 18, 18, 9, 1, 9, 2, 3, 30, 0, 12, 21, 38, 3, 27, 38, 2, 3, 2, 13, 3, 18, 34, 1, 5, 3, 28, 0, 1, 21, 14, 38, 1, 18, 40, 1, 2, 30, 65, 21, 34, 48, 64, 55, 45, 0, 49, 33, 60, 63, 3, 24, 5, 21, 2

Offset: 7

Amarnath Murthy, Jul 27 2005

			a(9) = 34 mod 7 = 6.

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Cf. A000045, A004090, A007953.

a:= n-> (f-> irem(f, add(i, i=convert(f, base, 10))))(combinat[fibonacci](n)):
seq(a(n), n=7..100);  # Alois P. Heinz, Jan 05 2022

Do[k = Fibonacci[n]; Print[Mod[k, Plus @@ IntegerDigits[k]]], {n, 7, 56}] (* Ryan Propper, Aug 14 2005 *)
Mod[#,Total[IntegerDigits[#]]]&/@Fibonacci[Range[7,70]] (* Harvey P. Dale, Dec 05 2015 *)

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

More terms from Ryan Propper, Aug 14 2005

More terms from Harvey P. Dale, Dec 05 2015

A112677 Sum of digits of the sum of the previous 4 terms.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 4, 7, 4, 4, 10, 7, 7, 10, 7, 4, 10, 4, 7, 7, 10, 10, 7, 7, 7, 4, 7, 7, 7, 7, 10, 4, 10, 4, 10, 10, 7, 4, 4, 7, 4, 10, 7, 10, 4, 4, 7, 7, 4, 4, 4, 10, 4, 4, 4, 4, 7, 10, 7, 10, 7, 7, 4, 10, 10, 4, 10, 7, 4, 7, 10, 10, 4, 4, 10, 10, 10, 7, 10, 10, 10, 10

Offset: 0

Jonathan Vos Post, Dec 30 2005

This is to the tetranacci sequence as A112661 is to the tribonacci and as A030132 is to Fibonacci. A000288 is the tetranacci sequence (A000078) but starting with values (1,1,1,1). Andrew Carmichael Post (andrewpost(AT)gmail.com) wrote the program that generated this sequence and showed that for any 4 initial integers a(0),a(1),a(2),a(3) the length of the cycle eventually entered is a factor of 312. For instance, starting with (6,6,6,6) continues in a cycle of length 1 since SOD(6+6+6+6) = SOD(24) = 6; and 1 divides 312. For the SOD(tribonacci) which is A112661, the length of any cycle eventually entered is a factor of 78.

All terms for n >= 4 are 4, 7, or 10. The sequence has period 78; the 78 terms after the initial 1,1,1,1 repeat forever. - Nathaniel Johnston, May 04 2011

			a(0)=a(1)=a(2)=a(3)=1.
a(4) = SOD(1+1+1+1) = SOD(4) = 4.
a(5) = SOD(1+1+1+4) = SOD(7) = 7.
a(10) = SOD(4+7+4+4) = SOD(19) = 10, note that we do not iterate SOD to reduce 10 to 1.

Cf. A000078, A000288, A004090, A007953, A010888, A030132, A112661.

A112677 := proc(n) option remember: if(n<=3)then return 1:fi: return add(d,d=convert(procname(n-1) + procname(n-2) + procname(n-3) + procname(n-4),base,10)): end: seq(A112677(n),n=0..100); # Nathaniel Johnston, May 04 2011

nxt[{a_,b_,c_,d_}]:={b,c,d,Total[IntegerDigits[a+b+c+d]]}; Transpose[ NestList[ nxt,{1,1,1,1},90]][[1]] (* or *) PadRight[{1,1,1,1},120,{10,10,10,10,4,7,4,7,4,4,10,7,7,10,7,4,10,4,7,7,10,10,7,7,7,4,7,7,7,7,10,4,10,4,10,10,7,4,4,7,4,10,7,10,4,4,7,7,4,4,4,10,4,4,4,4,7,10,7,10,7,7,4,10,10,4,10,7,4,7,10,10,4,4,10,10,10,7}](* Harvey P. Dale, Mar 05 2016 *)

a(0)=a(1)=a(2)=a(3)=1. a(n) = SumDigits(a(n-1) + a(n-2) + a(n-3) + a(n-4)).
a(n) = SumDigits(A000288(n)).
a(n) = A007953(a(n-1) + a(n-2) + a(n-3) + a(n-4)). - Nathaniel Johnston, May 04 2011

Name corrected by Nathaniel Johnston, May 04 2011

A139537 Digit sum of Fibonacci primes.

Original entry on oeis.org

2, 3, 5, 4, 17, 8, 22, 28, 23, 41, 37, 98, 116, 112, 343, 424, 388, 409, 473, 526, 527, 2678, 4208, 5102, 8587, 9023, 13576, 24040, 28898, 33787, 35218, 47719, 77842, 98257, 122543, 139256, 188837, 374174, 409118, 555355, 557992, 568246, 873941, 987592, 1210312

Offset: 1

Parthasarathy Nambi, Jun 09 2008

			514229 is a Fibonacci prime whose digit sum is 23.

James C. McMahon, Table of n, a(n) for n = 1..56
Ron Knott, List of Fibonacci numbers.

Cf. A007953, A005478, A001605.

Subsequence of A004090.

DigitSum/@Select[Fibonacci[Range[10000]],PrimeQ] (* James C. McMahon, Jun 30 2025 *)

a(n) = A007953(A005478(n)) and a(n) = A004090(A001605(n)). - Michel Marcus, Jun 30 2025

a(22)-a(34) and a(35) onwards (using A001605) from James C. McMahon, Jun 30 2025

cp's OEIS Frontend

A261587 Sum of sexagesimal digits of Fibonacci numbers in base-60 representation.

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A337448 The numbers k for which Fibonacci(k) are Niven numbers.

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Magma

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A112661 Sum of digits of sum of previous 3 terms.

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A068500 Sequence of Fibonacci numbers whose sum of decimal digits sets a new record.

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A139563 Fibonacci numbers whose digit sum is a Lucas number.

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A216699 Digital root of cubes of Fibonacci numbers.

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

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A110390 a(n) = F(n) mod s(n) where s(n) is the sum of the digits of the n-th Fibonacci number F(n).