A004090 -id:A004090 - cp's OEIS frontend

A164955 Sequence obtained from Fibonacci numbers by taking the factorials of each digit and summing.

1, 1, 1, 2, 6, 120, 40320, 7, 3, 30, 240, 403200, 49, 14, 10086, 722, 408240, 368041, 40466, 40346, 6600, 363626, 10083, 46202, 41790, 5283, 362896, 403946, 45369, 363029, 40354, 364353, 408250, 45632, 90843, 368788, 363040, 50548, 807128, 404792, 281, 41308

Offset: 0

Parthasarathy Nambi, Sep 01 2009

There seem to be very few primes in this sequence.

			a(30) = 8!+3!+2!+0!+4!+0! = 40354 because Fibonacci(30) = 832040.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A000045, A000142, A004090, A061602.

a:= n-> add(i!, i=convert((<<0|1>, <1|1>>^n)[1,2], base, 10)):
seq(a(n), n=0..42);  # Alois P. Heinz, Jul 09 2023

Total[IntegerDigits[#]!]&/@Fibonacci[Range[0,40]] (* Harvey P. Dale, May 03 2011 *)

a(n) = A061602(A000045(n)). - Alois P. Heinz, Jul 09 2023

Offset corrected and more terms from Alois P. Heinz, Jul 09 2023

A181361 Sum of cubes of digits of Fibonacci(n).

Original entry on oeis.org

0, 1, 1, 8, 27, 125, 512, 28, 9, 91, 250, 1241, 129, 62, 713, 217, 1584, 1198, 709, 578, 900, 1010, 689, 1204, 1035, 601, 793, 1523, 885, 935, 611, 1261, 1620, 1204, 1843, 1493, 981, 1397, 2726, 1897, 378, 1045, 2314, 1409, 1317, 1193, 1541, 1585, 2556, 2749

Offset: 0

Carmine Suriano, Oct 15 2010

			a(11) = 1241 since Fibonacci(11) = 89 and 8^3+9^3 = 512+729 = 1241.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000045, sum of digits A004090, sum of squared digits A181357.

f:= proc(n) local t; add(t^3, t = convert(combinat:-fibonacci(n),base,10)) end proc:
map(f, [$0..100]); # Robert Israel, Jul 31 2025

Total[IntegerDigits[#]^3]&/@Fibonacci[Range[50]] (* Harvey P. Dale, Mar 25 2012 *)

a(n) = A055012(A000045(n)). - Michel Marcus, Jul 18 2013

a(0) = 0 inserted by Robert Israel, Jul 31 2025

A244923 Numbers n such that the digit sum of Fibonacci(n) is equal to the digit sum of Lucas(n).

Original entry on oeis.org

1, 13, 61, 73, 97, 217, 349, 649, 937, 1477, 1513, 1729, 2005, 2077, 2209, 3265, 3649, 3889, 4093, 4609, 4945, 5497, 5749, 5929, 6109, 7309, 7441, 8041, 8389, 8821, 9925, 10525, 10669, 11605, 13201, 13345, 16021, 18529, 18649, 20293, 21481, 22573, 22729, 24169

Offset: 1

Michel Lagneau, Jul 08 2014

Numbers n such that A004090(n) = A139374(n).
Subsequence of A017533.
It seems that n is odd. The primes of the sequence are: 13, 61, 73, 97, 349, 937, 3889, 4093, 5749, 7309, 8389, 8821, 21481, 22573, 24169, ...
Fibonacci(j) == Lucas(j) (mod 9) iff j == 1 (mod 12), so all a(n) == 1 (mod 12). - Robert Israel, Jul 10 2014

			13 is in the sequence because Fibonacci(13) = 233, Lucas(13) = 521 and 2+3+3 = 5+2+1 = 8.

Vincenzo Librandi, Table of n, a(n) for n = 1..95

Cf. A000032, A000045, A004090, A017533, A139374.

lst={}; Table[If[Total[IntegerDigits[LucasL[n]]] == Total[IntegerDigits[Fibonacci[n]]], AppendTo[lst, n]], {n, 0, 25000}]; lst
Select[Range[25000],Total[IntegerDigits[Fibonacci[#]]]==Total[IntegerDigits[LucasL[#]]]&] (* Harvey P. Dale, Mar 31 2024 *)

A362358 Alternating sum of digits of the Fibonacci numbers, with a plus sign for the last digit.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 2, -1, -10, 0, 12, 1, 2, 3, 5, -3, 13, -1, 1, -11, 1, 12, 2, 3, 5, -3, 13, 10, 1, 0, 1, -10, 13, 3, -17, 19, -9, 10, 1, 0, 1, 12, 13, 3, -6, 8, 2, -1, -10, 0, 1, 12, -9, 3, 5, -3, -9, -23, 1, -22, 34, -10, 2

Offset: 0

Wolfdieter Lang, May 26 2023

a(n) mod 11 = F(n) mod 11 = A105955(n). This is the mod 11 rule applied to F(n) = A000045.

			F(17) = 1597, s(n) = 4 - 1 = 3, a(17) = 7 - 9 + 5 - 1 = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000045, A004090, A007887, A055017, A060384, A105955.

a[n_]:=Sum[(-1)^(IntegerLength[Fibonacci[n]]-i) Part[IntegerDigits[Fibonacci[n]],i],{i,IntegerLength[Fibonacci[n]]}]; Array[a,66,0] (* Stefano Spezia, May 27 2023 *)

a(n) = my(d=Vecrev(digits(fibonacci(n)))); sum(k=1, #d, (-1)^(k+1)*d[k]); \\ Michel Marcus, May 28 2023

Let [f_s(n), f_{s(n)-1}, ..., f_0] be the list of digits of F(n) = A000040(n) with s(n) = A060384(n) - 1, then a(n) = Sum_{j=0..s(n)} (-1)^j*f_j.

a(n) = A055017(A000045(n)), for n >= 0.

A112678 Sum of digits of previous 5 terms.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 9, 8, 6, 11, 12, 10, 11, 5, 13, 6, 9, 8, 5, 5, 6, 6, 3, 7, 9, 4, 11, 7, 11, 6, 12, 11, 11, 6, 10, 5, 7, 12, 4, 11, 12, 10, 13, 5, 6, 10, 8, 6, 8, 11, 7, 4, 9, 12, 7, 12, 8, 12, 6, 9, 11, 10, 12, 12, 9, 9, 7, 13, 5, 7, 5, 10, 4, 4, 3, 8, 11, 3, 11, 9, 6, 4, 6, 9, 7, 5, 4, 4, 11, 4

Offset: 0

Jonathan Vos Post, Dec 30 2005

This is to the pentanacci sequence A001591 as A112661 is to the tribonacci and as A030132 is to Fibonacci. A000322 is the pentanacci sequence (A001591) but starting with values (1,1,1,1,1). Andrew Carmichael Post (andrewpost(AT)gmail.com) wrote the program that generated this sequence and showed that for any 5 initial integers a(0),a(1),a(2),a(3),a(4) the length of the cycle eventually entered is a factor of 2184. For the SOD(teranacci) the limit cycle length is always a factor of 312. For the SOD(tribonacci) which is A112661, the length of any cycle eventually entered is a factor of 78.

			a(0)=a(1)=a(2)=a(3)=a(4)=1.
a(5) = SOD(1+1+1+1+1) = SOD(5) = 5.
a(6) = SOD(1+1+1+1+5) = SOD(9) = 9.
a(7) = SOD(1+1+1+5+9) = SOD(17) = 8.
a(8) = SOD(1+1+5+9+8) = SOD(24) = 6.
a(9) = SOD(1+5+9+8+6) = SOD(29) = 11, note that we do not iterate SOD to reduce 11 to 2.

Cf. A000322, A001591, A004090, A007953, A010888, A030132, A112661.

a(0)=a(1)=a(2)=a(3)=a(4)=1. a(n) = SumDigits(a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)). a(n) = SumDigits(A000322(n)).

A216754 Digital root of fourth power of Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9

Offset: 0

Ravi Bhandari, Sep 15 2012

This sequence is periodic with period 12. Also, the first (2n - 1) terms are symmetric about n-th term, where n = 6k, k = 1, 2, 3, ...

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

Cf. A004090, A216676, A216699.

Table[NestWhile[Total[IntegerDigits[#]] &, Fibonacci[n]^4, # > 9 &], {n, 0, 86}] (* T. D. Noe, Oct 15 2012 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9},108]] (* Ray Chandler, Aug 27 2015 *)

Extended by Ray Chandler, Aug 27 2015

A329192 Fibonacci numbers with arithmetic mean of digits an integer (sum of digits = a multiple of number of digits).

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 55, 144, 987, 6765, 10946, 9227465, 225851433717, 8944394323791464, 160500643816367088, 83621143489848422977, 59425114757512643212875125, 30960598847965113057878492344, 127127879743834334146972278486287885163

Offset: 1

José de Jesús Camacho Medina, Nov 07 2019

			55 is a term as the arithmetic mean of digits is an integer: (5+5)/2 = 5.
144 is a term as the arithmetic mean of digits is an integer: (1+4+4)/3 = 3.
6765 is a term as the arithmetic mean of digits is an integer: (6+7+6+5)/4 = 6.

Cf. A000045, A004090, A060384, A069709, A164947.

Mathematica
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a(n) = A000045(A164947(n+1)).

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

A164955 Sequence obtained from Fibonacci numbers by taking the factorials of each digit and summing.

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A181361 Sum of cubes of digits of Fibonacci(n).

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A244923 Numbers n such that the digit sum of Fibonacci(n) is equal to the digit sum of Lucas(n).

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A362358 Alternating sum of digits of the Fibonacci numbers, with a plus sign for the last digit.

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A112678 Sum of digits of previous 5 terms.

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A216754 Digital root of fourth power of Fibonacci numbers.

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A329192 Fibonacci numbers with arithmetic mean of digits an integer (sum of digits = a multiple of number of digits).

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

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