A010888 -id:A010888 - cp's OEIS frontend

A031346 Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.

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

Offset: 0

Eric W. Weisstein

			For n = 999: A007954(999) = 729, A007954(729) = 126, A007954(126) = 12 and A007954(12) = 2. The fourth iteration of "multiply digits" yields a single-digit number, so a(999) = 4. - _Felix Fröhlich_, Jul 17 2016

M. Gardner, Fractal Music, Hypercards and More Mathematical Recreations from Scientific American, Persistence of Numbers, pp. 120-1; 186-7, W. H. Freeman NY 1992.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 35.

T. D. Noe, Table of n, a(n) for n = 0..10000
Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.
Eric Brier, Christophe Clavier, Linda Gutsche and David Naccache, The Multiplicative Persistence Conjecture Is True for Odd Targets, arXiv:2110.04263 [math.NT], 2021.
M. R. Diamond, Multiplicative persistence base 10: some new null results.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A007954 (product of decimal digits of n).
Cf. A010888 (additive digital root of n).
Cf. A031286 (additive persistence of n).
Cf. A031347 (multiplicative digital root of n).
Cf. A263131 (ordinal transform).
Cf. A003001.

f:=func; a:=[]; for n in [0..100] do s:=0; k:=n; while k ge 10 do s:=s+1; k:=f(k); end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 12 2020

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A031346 := proc(n) local k,m: k:=0:m:=n: while(length(m)>1)do m:=A007954(m):k:=k+1: od: return k: end: seq(A031346(n),n=0..100); # Nathaniel Johnston, May 04 2011

Table[Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>=10&]],{n,0,100}]-1 (* Harvey P. Dale, Aug 27 2016 *)

a007954(n) = my(d=digits(n)); prod(i=1, #d, d[i])
a(n) = my(k=n, i=0); while(#Str(k) > 1, k=a007954(k); i++); i \\ Felix Fröhlich, Jul 17 2016

from operator import mul
from functools import reduce
def A031346(n):
    mp = 0
    while n > 9:
        n = reduce(mul, (int(d) for d in str(n)))
        mp += 1
    return mp
# Chai Wah Wu, Aug 23 2014

Probably bounded, see A003001. - Charles R Greathouse IV, Nov 15 2022

A038194 Iterated sum-of-digits of n-th prime; or digital root of n-th prime; or n-th prime modulo 9.

Original entry on oeis.org

2, 3, 5, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 8, 5, 7, 4, 8, 1, 7, 2, 8, 7, 2, 4, 8, 1, 5, 1, 5, 2, 4, 5, 7, 4, 1, 5, 2, 8, 1, 2, 4, 8, 1, 4, 7, 2, 4, 8, 5, 7, 8, 5, 2, 8, 1, 7, 2, 4, 5, 1, 5, 7, 2, 7, 4, 5, 7, 2, 8, 7, 4, 1, 5, 2, 1, 5, 4, 5, 7, 8, 1, 7, 2, 8

Offset: 1

Den Roussel (DenRoussel(AT)webtv.net) and Clark Kimberling

Integers with iterated sum-of-digits 3, 6 or 9 are divisible by 3, so 3 is the only prime with iterated sum-of-digits 3 and there are no primes with iterated sum-of-digits 6 or 9.
The remaining values are very evenly distributed: these are the number of appearances in the first 1007933 primes: 1:167878; 2:168079; 4:167984; 5:168027; 7:167906; 8:168058. - Carmine Suriano, Jun 22 2015
Asymptotically, the ratios (number of primes <= n and == i mod 9)/(number of primes <= n and == j mod 9) go to 1 as n -> infinity for all i,j in {1,2,4,5,7,8} by the Prime Number Theorem for Arithmetic Progressions. For more detailed analysis, see the Granville-Martin link. - Robert Israel, Jul 08 2015

			Prime(5) = 11, 1 + 1 = 2 hence a(5) = 2.
a(297)=7 because the 297th prime is 1951 and 1+9+5+1 = 16 -> 1+6 = 7.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004

Cf. A007605, A010888, A061237 - A061242, A139413, A153110.

a038194 = flip mod 9 . a000040  -- Reinhard Zumkeller, Dec 10 2014

[p mod 9: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

A038194 := proc(n) return ithprime(n) mod 9: end: seq(A038194(n), n=1..100); # Nathaniel Johnston, May 04 2011

Table[Mod[Prime[n], 9], {n, 200}]
Mod[Prime[Range[100]], 9] (* Vincenzo Librandi, May 06 2014 *)

PARI
```
forprime(p=2,600,print1(p%9,","))
```

a(n) = A010888(A000040(n)).

Sum_k={1..n} a(k) ~ (9/2)*n. - Amiram Eldar, Dec 11 2024

Edited by Klaus Brockhaus, Feb 16 2002

Edited at the suggestion of R. J. Mathar by N. J. A. Sloane, May 14 2008

A017173 a(n) = 9*n + 1.

Original entry on oeis.org

1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478

Offset: 0

N. J. A. Sloane

Also all the numbers with digital root 1; A010888(a(n)) = 1. - Rick L. Shepherd, Jan 12 2009
A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - Reinhard Zumkeller, Feb 05 2009
If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093644 ((9,1) Pascal, column m=1).
Cf. A010701, A010888, A016777, A116371, A147296, A156144, A156145.
Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).

a017173 = (+ 1) . (* 9)
a017173_list = [1, 10 ..]  -- Reinhard Zumkeller, Feb 04 2014

Range[1, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{1,10},60] (* Harvey P. Dale, Dec 27 2014 *)

forstep(n=1,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

[i+1 for i in range(480) if gcd(i,9) == 9] # Zerinvary Lajos, May 20 2009

G.f.: (1 + 8*x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - Vincenzo Librandi, Aug 01 2010
E.g.f.: exp(x)*(1 + 9*x). - Stefano Spezia, Apr 20 2023
a(n) = A016777(3*n). - Elmo R. Oliveira, Apr 12 2025

A027687 4-perfect (quadruply-perfect or sous-triple) numbers: sum of divisors of n is 4n.

Original entry on oeis.org

30240, 32760, 2178540, 23569920, 45532800, 142990848, 1379454720, 43861478400, 66433720320, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408

Offset: 1

Jean-Yves Perrier (nperrj(AT)ascom.ch)

nonn

It is conjectured that there are only finitely many terms. - N. J. A. Sloane, Jul 22 2012
Odd perfect number (unlikely to exist) and infinitely many Mersenne primes will make the sequence infinite - take the product of the OPN and coprime EPNs.
Conjecture: A010888(a(n)) divides a(n). Tested for n up to 36 incl. - Ivan N. Ianakiev, Oct 31 2013
From Farideh Firoozbakht, Dec 26 2014: (Start)
Theorem: If k>1 and p=a(n)/2^(k-2)+1 is prime then for each positive integer m, 2^(k-1)*p^m is a solution to the equation sigma(phi(x))=2*x-2^k, which implies the equation has infinitely many solutions.
Proof: sigma(phi(2^(k-1)*p^m)) = sigma(2^(k-2)*(p-1)*p^(m-1)) = sigma(2^(k-2)*(p-1))*sigma(p^(m-1)) = sigma(a(n))*(p^m-1)/(p-1) = 4*a(n)*(p^m-1)/(p-1) = 2^k*(p^m-1) = 2*(2^(k-1)*p^m)-2^k.
It seems that for all such equations there exist such an infinite set of solutions. So I conjecture that the sequence is infinite! (End)
If 3 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k)=4. - Michel Marcus, Nov 22 2015

			From _Daniel Forgues_, May 09 2010: (Start)
30240 = 2^5*3^3*5*7
sigma(30240) = (2^6-1)/1*(3^4-1)/2*(5^2-1)/4*(7^2-1)/6
= (63)*(40)*(6)*(8)
= (7*3^2)*(2^3*5)*(2*3)*(2^3)
= 2^7*3^3*5*7
= (2^2) * (2^5*3^3*5*7)
= 4 * 30240 (End)

R. K. Guy, Unsolved Problems in Number Theory, B2.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 143.

T. D. Noe, Table of n, a(n) for n=1..36 (complete sequence from Flammenkamp)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Kevin A. Broughan and Qizhi Zhou, Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4, JIS 13 (2010) 10.1.5
Seth Colbert-Pollack, Judy Holdener, Emily Rachfal, and Yanqi Xu, A DIY Project: Construct Your Own Multiply Perfect Number!, Math Horizons, Vol. 28, pp. 20-23, February 2021.
Farideh Firoozbakht and Maxmilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Fred Helenius, Link to Glossary and Lists
Walter Nissen, Abundancy : Some Resources
Eric Weisstein's World of Mathematics, Multiperfect Number.
Eric Weisstein's World of Mathematics, Sous-Triple.
Wikipedia, Multiply perfect number

Cf. A000396, A005820, A007539, A046060, A046061.

AbundantQ[n_]:=DivisorSigma[1, n]==4*n;a={};Do[If[AbundantQ[n], AppendTo[a, n]], {n, 10^6}];a (* Vladimir Joseph Stephan Orlovsky, Aug 16 2008 *)

4 more terms from Labos Elemer

A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n(n+1)(5*n-2)/6.

Original entry on oeis.org

0, 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, 2380, 2920, 3536, 4233, 5016, 5890, 6860, 7931, 9108, 10396, 11800, 13325, 14976, 16758, 18676, 20735, 22940, 25296, 27808, 30481, 33320, 36330, 39516, 42883, 46436, 50180, 54120

Offset: 0

N. J. A. Sloane

The partial sums of A000566. - R. J. Mathar, Mar 19 2008
A002413(n + 1) is the number of 4-tuples (w, x, y, z) having all terms in {0, ..., n} and w = floor((x + y + z)/2).  - Clark Kimberling, May 28 2012
From Ant King, Oct 25 2012: (Start)
For n > 0, the digital roots of this sequence A010888(A002413(n)) form the purely periodic 27-cycle {1, 8, 8, 6, 7, 7, 2, 6, 6, 7, 5, 5, 3, 4, 4, 8, 3, 3, 4, 2, 2, 9, 1, 1, 5, 9, 9}.
For n > 0, the units' digits of this sequence A010879(A002413(n)) form the purely periodic 20-cycle {1, 8, 6, 0, 5, 6, 8, 6, 5, 0, 6, 8, 1, 0, 0, 6, 3, 6, 0, 0}.
(End)

			For n=7, a(7) = 7*1 + 6*6 + 5*11 + 4*16 + 3*21 + 2*26 + 1*31 = 308. - _Bruno Berselli_, Feb 10 2014

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A093562 ((5, 1) Pascal, column m = 3).

Cf. similar sequences listed in A237616.

[n*(n + 1)*(5*n - 2)/6: n in [0..50]]; // G. C. Greubel, Nov 04 2017

A002413:=n->n*(n+1)*(5*n-2)/6: seq(A002413(n), n=0..60); # Wesley Ivan Hurt, Apr 14 2017

LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 60}, 40] (* Ant King, Oct 25 2012 *)
Table[(5n^3 + 3n^2 - 2n)/6, {n, 0, 39}] (* Alonso del Arte, Oct 25 2012 *)

A002413(n):=n*(n+1)*(5*n-2)/6$ makelist(A002413(n),n,0,20); /* Martin Ettl, Dec 12 2012 */

a(n)=n*(n+1)*(5*n-2)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(n + 1)*(5*n - 2)/6.
G.f.: x*(1 + 4*x)/(1 - x)^4. [Suggested by Simon Plouffe in his 1992 dissertation.]
From Ant King, Oct 25 2012: (Start)
a(n) = a(n - 1) + n*(5*n - 3)/2.
a(n) = 3*a(n - 1) - 3*a(n - 2) + a(n - 3) + 5.
a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4)
a(n) = (n + 1)*(2*A000566(n) + n)/6 = (5*n - 2)*A000217(n)/3.
a(n) = A000292(n) + 4*A000292(n - 1)
a(n) = A002412(n) + A000292(n - 1)
a(n) = A000217(n) + 5*A000292(n - 1)
a(n) = binomial(n + 2, 3) + 4*binomial(n + 1, 3) = (5*n - 2) * binomial(n + 1, 2)/3.
Sum_{n >= 1} 1/a(n) = 15*(log(3125) + sqrt(5)*log((3 - sqrt(5))/2) - 2*Pi*sqrt(5*(5 - 2*sqrt(5)))/5 - 8/5)/28 = 1.207293...
(End)
a(n) = Sum_{i=0..n-1} (n-i)*(5*i+1). - Bruno Berselli, Feb 10 2014
a(n) = A080851(5,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: x*(6 + 18*x + 5*x^2)*exp(x)/6. - Ilya Gutkovskiy, May 12 2017
a(n) = Sum_{i=0..n-1} (n+2*i)*(n-i). - Leonid Bedratyuk, Jul 09 2024

More terms from James Sellers, Dec 23 1999

a(0)=0 prepended by Max Alekseyev, Nov 23 2011

A010878 a(n) = n mod 9.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Periodic with period of length 9. The digital root of n (A010888) is a very similar sequence.

The rightmost digit in the base-9 representation of n. Also, the equivalent value of the two rightmost digits in the base-3 representation of n. - Hieronymus Fischer, Jun 11 2007

Ely Golden, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Partial sums: A130487. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486.

a010878 = (`mod` 9)
a010878_list = cycle [0..8]  -- Reinhard Zumkeller, Jan 09 2013

A010878 := proc(n)
    modp(n,9) ;
end proc:
seq(A010878(n),n=0..100) ; # R. J. Mathar, Sep 09 2015

Array[Mod[#, 9]&, 105, 0] (* Jean-François Alcover, Jan 30 2018 *)
PadRight[{},120,Range[0,8]] (* Harvey P. Dale, Dec 19 2018 *)

a(n)=n%9 \\ Charles R Greathouse IV, Sep 24 2015

Complex representation: a(n)=(1/9)*(1-r^n)*sum{1<=k<9, k*product{1<=m<9,m<>k, (1-r^(n-m))}} where r=exp(2*pi/9*i) and i=sqrt(-1). Trigonometric representation: a(n)=(256/9)^2*(sin(n*pi/9))^2*sum{1<=k<9, k*product{1<=m<9,m<>k, (sin((n-m)*pi/9))^2}}. G.f.: g(x)=(sum{1<=k<9, k*x^k})/(1-x^9). Also: g(x)=x(8x^9-9x^8+1)/((1-x^9)(1-x)^2). - Hieronymus Fischer, May 31 2007
a(n) = n mod 3 + 3*(floor(n/3)mod 3) = A010872(n) + 3*A010872(A002264(n)). - Hieronymus Fischer, Jun 11 2007
a(n) = floor(12345678/999999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = floor(1513361/96855122*9^(n+1)) mod 9. - Hieronymus Fischer, Jan 04 2013

A017209 a(n) = 9*n + 4.

Original entry on oeis.org

4, 13, 22, 31, 40, 49, 58, 67, 76, 85, 94, 103, 112, 121, 130, 139, 148, 157, 166, 175, 184, 193, 202, 211, 220, 229, 238, 247, 256, 265, 274, 283, 292, 301, 310, 319, 328, 337, 346, 355, 364, 373, 382, 391, 400, 409, 418, 427, 436, 445, 454, 463, 472, 481

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 4. - L. Edson Jeffery, Nov 26 2016

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D5.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A010888, A017185, A017197.

[9*n+4: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[4, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

forstep(n=4,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

G.f.: (4 + 5*x)/(x - 1)^2. - R. J. Mathar, Jul 14 2016
A010888(a(n)) = 4. - L. Edson Jeffery, Nov 26 2016
E.g.f.: exp(x)*(4 + 9*x). - Stefano Spezia, Dec 25 2022

A004090 Sum of digits of Fibonacci numbers.

Original entry on oeis.org

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

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

Original entry on oeis.org

1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5

Offset: 0

Paul Curtz, Dec 19 2008

Digital root of 2^n.
A regular version of Pitoun's sequence: a(n) = A029898(n+1).
Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
This sequence and its (again period 6) repeated differences produce the table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4, -1, -2, -4,  1,  2,  4, -1, -2, ...
    1,  2, -5, -1, -2,  5,  1,  2, -5, -1, -2, ...
    1, -7,  4, -1,  7, -4,  1, -7,  4, -1,  7, ...
   -8, 11, -5,  8,-11,  5, -8, 11, -5,  8,-11, ...
   19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
  -35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
   64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
If each entry of this table is read modulo 9 we obtain the very regular table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
Also the decimal expansion of the constant 125/1001. - R. J. Mathar, Jan 23 2009
Digital root of the powers of any number congruent to 2 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. A004000, A007953, A010734, A010888, A029898, A030132, A082365, A145389, A189510.
Cf. digital roots of powers of c mod 9: c = 4, A100402; c = 5, A070366; c = 7, A070403; c = 8, A010689.
Cf. A020806, A070366, A141425, A153110, A153990, A154127, A154687, A154815.

&cat [[1, 2, 4, 8, 7, 5]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* Paul Curtz, Dec 19 2008 *)
Table[Mod[2^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)

a(n)=lift(Mod(2,9)^n) \\ Charles R Greathouse IV, Apr 21 2015

a(n) + a(n+3) = 9 = A010734(n).
G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - R. J. Mathar, Jan 23 2009
a(n) = A082365(n) mod 9. - Paul Curtz, Mar 31 2009
a(n) = -1/2*cos(Pi*n) - 3*cos(1/3*Pi*n) - 3^(1/2)*sin(1/3*Pi*n) + 9/2. - Leonid Bedratyuk, May 13 2012
a(n) = A010888(A004000(n+1)). - Ivan N. Ianakiev, Nov 27 2014
From Wesley Ivan Hurt, Apr 20 2015: (Start)
a(n) = a(n-6) for n>5.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
a(n) = 2^n mod 9. - Nikita Sadkov, Oct 06 2018
From Stefano Spezia, Mar 20 2025: (Start)
E.g.f.: 4*cosh(x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 5*sinh(x).
a(n) = A007953(2*a(n-1)) = A010888(2*a(n-1)). (End)

Edited by R. J. Mathar, Apr 09 2009

A031286 Additive persistence: number of summations of digits needed to obtain a single digit (the additive digital root).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Eric W. Weisstein

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Antonios Meimaris, On the additive persistence of a number in base p, Preprint, 2015.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Additive Persistence

Cf. A010888 (additive digital root of n).
Cf. A031347 (multiplicative digital root of n).
Cf. A031346 (multiplicative persistence of n).
Cf. also A006050, A045646.
Cf. Numbers with additive persistence k: A304366 (k=1), A304367 (k=2), A304368 (k=3), A304373 (k=4). - Jaroslav Krizek, May 28 2018

read("transforms") ;
A031286 := proc(n)
    local a,nper;
    nper := n ;
    a := 0 ;
    while nper > 9 do
        nper := digsum(nper) ;
        a := a+1 ;
    end do:
    a ;
end proc:
seq(A031286(n),n=0..80) ; # R. J. Mathar, Jan 02 2018

lst = {}; Do[s = 0; While[n > 9, s++; n = Plus @@ IntegerDigits[n]]; AppendTo[lst, s], {n, 0, 98}]; lst (* Arkadiusz Wesolowski, Oct 17 2012 *)

dsum(n)=my(s);while(n,s+=n%10;n\=10);s
a(n)=my(s);while(n>9,s++;n=dsum(n));s \\ Charles R Greathouse IV, Sep 13 2012

def A031286(n):
    ap = 0
    while n > 9:
        n = sum(int(d) for d in str(n))
        ap += 1
    return ap
# Chai Wah Wu, Aug 23 2014

Corrected by Reinhard Zumkeller, Feb 05 2009

cp's OEIS Frontend

A031346 Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A038194 Iterated sum-of-digits of n-th prime; or digital root of n-th prime; or n-th prime modulo 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A017173 a(n) = 9*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Sage

Formula

A027687 4-perfect (quadruply-perfect or sous-triple) numbers: sum of divisors of n is 4n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

Extensions

A010878 a(n) = n mod 9.

Views

Author

Keywords

Comments

Links

A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n(n+1)(5*n-2)/6.