A112765 -id:A112765 - cp's OEIS frontend

A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

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

Offset: 1

John Tromp, Dec 11 1996

This sequence is an exception to my usual rule that when every other term of a sequence is 0 then those 0's should be omitted. In this case we would get A001511. - N. J. A. Sloane
To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre, Mar 06 2003
The sequence is invariant under the following two transformations: increment every element by one (1, 2, 1, 3, 1, 2, 1, 4, ...), put a zero in front and between adjacent elements (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...). The intermediate result is A001511. - Ralf Hinze (ralf(AT)informatik.uni-bonn.de), Aug 26 2003
Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), ..., starting from a(1) = 0. - Philippe Deléham, Mar 15 2004
Fixed point of the morphism 0->010, 1->2, 2->3, ..., n->(n+1), .... - Joerg Arndt, Apr 29 2014
a(n) is also the number of times to repeat a step on an even number in the hailstone sequence referenced in the Collatz conjecture. - Alex T. Flood (whiteangelsgrace(AT)gmail.com), Sep 22 2006
Let F(n) be the n-th Fermat number (A000215). Then F(a(r-1)) divides F(n)+2^k for r = k mod 2^n and r != 1. - T. D. Noe, Jul 12 2007
The following relation holds: 2^A007814(n)*(2*A025480(n-1)+1) = A001477(n) = n. (See functions hd, tl and cons in [Paul Tarau 2009].)
a(n) is the number of 0's at the end of n when n is written in base 2.
a(n+1) is the number of 1's at the end of n when n is written in base 2. - M. F. Hasler, Aug 25 2012
Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 0). That is, A003188(n) XOR A003188(n+1) == 2^A007814(n). - Russ Cox, Dec 04 2010
The sequence is squarefree (in the sense of not containing any subsequence of the form XX) [Allouche and Shallit]. Of course it contains individual terms that are squares (such as 4). - Comment expanded by N. J. A. Sloane, Jan 28 2019
a(n) is the number of zero coefficients in the n-th Stern polynomial, A125184. - T. D. Noe, Mar 01 2011
Lemma: For n < m with r = a(n) = a(m) there exists n < k < m with a(k) > r. Proof: We have n=b2^r and m=c2^r with b < c both odd; choose an even i between them; now a(i2^r) > r and n < i2^r < m. QED. Corollary: Every finite run of consecutive integers has a unique maximum 2-adic valuation. - Jason Kimberley, Sep 09 2011
a(n-2) is the 2-adic valuation of A000166(n) for n >= 2. - Joerg Arndt, Sep 06 2014
a(n) = number of 1's in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} p_j-th prime (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24)=3; indeed, the partition having Heinz number 24 = 2*2*2*3  is [1,1,1,2]. - Emeric Deutsch, Jun 04 2015
a(n+1) is the difference between the two largest parts in the integer partition having viabin number n (0 is assumed to be a part). Example: a(20) = 2. Indeed, we have 19 = 10011_2, leading to the Ferrers board of the partition [3,1,1]. For the definition of viabin number see the comment in A290253. - Emeric Deutsch, Aug 24 2017
Apart from being squarefree, as noted above, the sequence has the property that every consecutive subsequence contains at least one number an odd number of times. - Jon Richfield, Dec 20 2018
a(n+1) is the 2-adic valuation of Sum_{e=0..n} u^e = (1 + u + u^2 + ... + u^n), for any u of the form 4k+1 (A016813). - Antti Karttunen, Aug 15 2020
{a(n)} represents the "first black hat" strategy for the game of countably infinitely many hats, with a probability of success of 1/3; cf. the Numberphile link below. - Frederic Ruget, Jun 14 2021
a(n) is the least nonnegative integer k for which there does not exist i+j=n and a(i)=a(j)=k (cf. A322523). - Rémy Sigrist and Jianing Song, Aug 23 2022

			2^3 divides 24, so a(24)=3.
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
  0;
  1,0;
  2,0,1,0;
  3,0,1,0,2,0,1,0;
  4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,...
(End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 27.
K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014.
K. Atanassov, On Some of Smarandache's Problems
Alain Connes, Caterina Consani, and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], 2023.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, (2009) Discrete Math., 309 (2009), 6245-6254.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
M. Hassani, Equations and inequalities involving v_p(n!), J. Inequ. Pure Appl. Math. 6 (2005) vol. 2, #29.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 61. Book's website
R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.3.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.
Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Nicolas Mallet, Trial for a proof of the Syracuse conjecture, arXiv preprint arXiv:1507.05039 [math.GM], 2015.
S. Mazzanti, Plain Bases for Classes of Primitive Recursive Functions, Mathematical Logic Quarterly, 48 (2002).
Matthew Andres Moreno, Luis Zaman, and Emily Dolson, Structured Downsampling for Fast, Memory-efficient Curation of Online Data Streams, arXiv:2409.06199 [cs.DS], 2024. See p. 5.
Sascha Mücke, Coding Nuggets Faster QUBO Brute-Force Solving, TU Dortmund Univ. (Germany 2023).
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Numberphile, Hat Problems - Numberphile
Giovanni Pighizzini, Limited Automata: Properties, Complexity and Variants, International Conference on Descriptional Complexity of Formal Systems (DCFS 2019) Descriptional Complexity of Formal Systems, Lecture Notes in Computer Science (LNCS, Vol. 11612) Springer, Cham, 57-73.
Simon Plouffe, On the values of the functions ... [zeta and Gamma] ..., arXiv preprint arXiv:1310.7195 [math.NT], 2013.
A. Postnikov (MIT) and B. Sagan, What power of two divides a weighted Catalan number?, arXiv:math/0601339 [math.CO], 2006.
Lara Pudwell and Eric Rowland, Avoiding fractional powers over the natural numbers, arXiv:1510.02807 [math.CO] (2015). Also Electronic Journal of Combinatorics, Volume 25(2) (2018), #P2.27. See Section 2.
Ville Salo, Decidability and Universality of Quasiminimal Subshifts, arXiv preprint arXiv:1411.6644 [math.DS], 2014.
Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2014.
F. Smarandache, Only Problems, Not Solutions!.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Paul Tarau, A Groupoid of Isomorphic Data Transformations. Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.
P. M. B. Vitanyi, An optimal simulation of counter machines, SIAM J. Comput, 14:1(1985), 1-33.
Eric Weisstein's World of Mathematics, Binary, Binary Carry Sequence, and Double-Free Set.
Wikipedia, P-adic order.
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to binary expansion of n

Cf. A011371 (partial sums), A094267 (first differences), A001511 (bisection), A346070 (mod 4).
Bisection of A050605 and |A088705|. Pairwise sums are A050603 and A136480. Difference of A285406 and A281264.
This is Guy Steele's sequence GS(1, 4) (see A135416). Cf. A053398(1,n). Column/row 1 of table A050602.
Cf. A007949 (3-adic), A235127 (4-adic), A112765 (5-adic), A122841 (6-adic), A214411 (7-adic), A244413 (8-adic), A122840 (10-adic).
Cf. A000079, A002487, A006519, A051064, A055457, A063787, A215366, A220466.
Cf. A086463 (Dgf at s=2).

a007814 n = if m == 0 then 1 + a007814 n' else 0
            where (n', m) = divMod n 2
-- Reinhard Zumkeller, Jul 05 2013, May 14 2011, Apr 08 2011

a007814 n | odd n = 0 | otherwise = 1 + a007814 (n `div` 2)
--  Walt Rorie-Baety, Mar 22 2013

[Valuation(n, 2): n in [1..120]]; // Bruno Berselli, Aug 05 2013

ord := proc(n) local i,j; if n=0 then return 0; fi; i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end proc: seq(ord(n), n=1..111);
A007814 := n -> padic[ordp](n,2): seq(A007814(n), n=1..111); # Peter Luschny, Nov 26 2010

Table[IntegerExponent[n, 2], {n, 64}] (* Eric W. Weisstein *)
IntegerExponent[Range[64], 2] (* Eric W. Weisstein, Feb 01 2024 *)
p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ]
DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on the same concept: Floor[Log[2, BitXor[x, x - 1]]] (* Jaume Simon Gispert (jaume(AT)nuem.com), Aug 29 2004 *)
Nest[Join[ #, ReplacePart[ #, Length[ # ] -> Last[ # ] + 1]] &, {0, 1}, 5] (* N. J. Gunther, May 23 2009 *)
Nest[ Flatten[# /. a_Integer -> {0, a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 17 2011 *)

PARI
```
A007814(n)=valuation(n,2);
```

import math
def a(n): return int(math.log(n - (n & n - 1), 2)) # Indranil Ghosh, Apr 18 2017

def A007814(n): return (~n & n-1).bit_length() # Chai Wah Wu, Jul 01 2022

sapply(1:100,function(x) sum(gmp::factorize(x)==2)) # Christian N. K. Anderson, Jun 20 2013

(define (A007814 n) (let loop ((n n) (e 0)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

a(n) = A001511(n) - 1.
a(2*n) = A050603(2*n) = A001511(n).
a(n) = A091090(n-1) + A036987(n-1) - 1.
a(n) = 0 if n is odd, otherwise 1 + a(n/2). - Reinhard Zumkeller, Aug 11 2001
Sum_{k=1..n} a(k) = n - A000120(n). - Benoit Cloitre, Oct 19 2002
G.f.: A(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Apr 10 2002
G.f. A(x) satisfies A(x) = A(x^2) + x^2/(1-x^2). A(x) = B(x^2) = B(x) - x/(1-x), where B(x) is the g.f. for A001151. - Franklin T. Adams-Watters, Feb 09 2006
Totally additive with a(p) = 1 if p = 2, 0 otherwise.
Dirichlet g.f.: zeta(s)/(2^s-1). - Ralf Stephan, Jun 17 2007
Define 0 <= k <= 2^n - 1; binary: k = b(0) + 2*b(1) + 4*b(2) + ... + 2^(n-1)*b(n-1); where b(x) are 0 or 1 for 0 <= x <= n - 1; define c(x) = 1 - b(x) for 0 <= x <= n - 1; Then: a(k) = c(0) + c(0)*c(1) + c(0)*c(1)*c(2) + ... + c(0)*c(1)...c(n-1); a(k+1) = b(0) + b(0)*b(1) + b(0)*b(1)*b(2) + ... + b(0)*b(1)...b(n-1). - Arie Werksma (werksma(AT)tiscali.nl), May 10 2008
a(n) = floor(A002487(n - 1) / A002487(n)). - Reikku Kulon, Oct 05 2008
Sum_{k=1..n} (-1)^A000120(n-k)*a(k) = (-1)^(A000120(n)-1)*(A000120(n) - A000035(n)). - Vladimir Shevelev, Mar 17 2009
a(A001147(n) + A057077(n-1)) = a(2*n). - Vladimir Shevelev, Mar 21 2009
For n>=1, a(A004760(n+1)) = a(n). - Vladimir Shevelev, Apr 15 2009
2^(a(n)) = A006519(n). - Philippe Deléham, Apr 22 2009
a(n) = A063787(n) - A000120(n). - Gary W. Adamson, Jun 04 2009
a(C(n,k)) = A000120(k) + A000120(n-k) - A000120(n). - Vladimir Shevelev, Jul 19 2009
a(n!) = n - A000120(n). - Vladimir Shevelev, Jul 20 2009
v_{2}(n) = Sum_{r>=1} (r / 2^(r+1)) Sum_{k=0..2^(r+1)-1} e^(2(k*Pi*i(n+2^r))/(2^(r+1))). - A. Neves, Sep 28 2010, corrected Oct 04 2010
a(n) mod 2 = A096268(n-1). - Robert G. Wilson v, Jan 18 2012
a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7; a(n) = (A037227(n)-1)/2. - Reinhard Zumkeller, Jun 30 2012
a((2*n-1)*2^p) = p, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 04 2013
a(n) = A067255(n,1). - Reinhard Zumkeller, Jun 11 2013
a(n) = log_2(n - (n AND n-1)). - Gary Detlefs, Jun 13 2014
a(n) = 1 + A000120(n-1) - A000120(n), where A000120 is the Hamming weight function. - Stanislav Sykora, Jul 14 2014
A053398(n,k) = a(A003986(n-1,k-1)+1); a(n) = A053398(n,1) = A053398(n,n) = A053398(2*n-1,n) = Min_{k=1..n} A053398(n,k). - Reinhard Zumkeller, Aug 04 2014
a((2*x-1)*2^n) = a((2*y-1)*2^n) for positive n, x and y. - Juri-Stepan Gerasimov, Aug 04 2016
a(n) = A285406(n) - A281264(n). - Ralf Steiner, Apr 18 2017
a(n) = A000005(n)/(A000005(2*n) - A000005(n)) - 1. - conjectured by Velin Yanev, Jun 30 2017, proved by Nicholas Stearns, Sep 11 2017
Equivalently to above formula, a(n) = A183063(n) / A001227(n), i.e., a(n) is the number of even divisors of n divided by number of odd divisors of n. - Franklin T. Adams-Watters, Oct 31 2018
a(n)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Feb 16 2019
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Jul 11 2020
a(n) = 2*Sum_{j=1..floor(log_2(n))} frac(binomial(n, 2^j)*2^(j-1)/n). - Dario T. de Castro, Jul 08 2022
a(n) = A070939(n) - A070939(A030101(n)). - Andrew T. Porter, Dec 16 2022
a(n) = floor((gcd(n, 2^n)^(n+1) mod (2^(n+1)-1)^2)/(2^(n+1)-1)) (see Lemma 3.4 from Mazzanti's 2002 article). - Lorenzo Sauras Altuzarra, Mar 10 2024
a(n) = 1 - A088705(n). - Chai Wah Wu, Sep 18 2024

Formula index adapted to the offset of A025480 by R. J. Mathar, Jul 20 2010

Edited by Ralf Stephan, Feb 08 2014

A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n.

Original entry on oeis.org

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

Offset: 1

R. Muller

Obeys the general recurrences for p-adic valuation discussed in A214411. - Redjan Shabani, Jul 17 2012
Lexicographically earliest cubefree sequence, which also (conjecturally) appears in the construction of the lexicographically earliest cubefree {0,1}-sequence A282317, cf. Example section of A286940. - M. F. Hasler, May 21 2017
The sequence is invariant under the "lower trim" operator: remove all zeros, and subtract one from each remaining term. - Franklin T. Adams-Watters, May 25 2017

F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Atanassov, On the 61-st, 62-nd and the 63-rd Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4 (1998), No. 4, 175-182.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
F. Smarandache, Only Problems, Not Solutions!.
M. Vassilev-Missana and K. Atanassov, Some Representations related to n!, Notes on Number Theory and Discrete Mathematics, Vol. 4 (1998), No. 4, 148-153.
Index entries for sequences that are fixed points of mappings

Partial sums give A054861.
Cf. A000005, A038500, A080278, A001511, A122841, A007814, A112765, A253786.
One less than A051064.

a007949 n = if m > 0 then 0 else 1 + a007949 n'
            where (n', m) = divMod n 3
-- Reinhard Zumkeller, Jun 23 2013, May 14 2011

% Input:
%  n: an integer
% Output:
%  m: max power of 3 such that 3^m divides n
%  M: 1-by-K matrix where M(i) is the max power of 3 such that 3^M(i) divides n
function [m,M] = Omega3(n)
  M = NaN*zeros(1,n);
  M(1)=0; M(2)=0; M(3)=0;
    for k=4:n
      if M(k-3)~=0
        M(k)=M(k-k/3)+1;
      else
        M(k)=0;
      end
    end
    m=M(end);
end
% Redjan Shabani, Jul 17 2012

[Valuation(n, 3): n in [1..110]]; // Bruno Berselli, Aug 05 2013

A007949 := proc(n) option remember; if n mod 3 > 0 then 0 else procname(n/3)+1; fi; end;
# alternative by R. J. Mathar, Mar 29 2017
A007949 := proc(n)
    padic[ordp](n,3) ;
end proc:

p=3; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 81 ]
Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 0, 1}, 1 -> {0, 0, 2}, 2 -> {0, 0, 3}, 3 -> {0, 0, 4}}) ]}], {0}, 5] (* Robert G. Wilson v, Mar 03 2005 *)
IntegerExponent[Range[200], 3] (* Zak Seidov, Apr 15 2010 *)
Table[If[Mod[n, 3] > 0, 0, 1 + b[n/3]], {n, 200}] (* Zak Seidov, Apr 15 2010 *)

PARI
```
a(n)=valuation(n,3)
```

def a(n):
    k = 0
    while n > 0 and n%3 == 0: n //= 3; k += 1
    return k
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 06 2021

[valuation(n, 3) for n in (1..106)]  # Peter Luschny, Nov 16 2012

(define (A007949 n) (let loop ((n n) (k 0)) (cond ((not (zero? (modulo n 3))) k) (else (loop (/ n 3) (+ 1 k)))))) ;; Antti Karttunen, Oct 06 2017

a(n) = 0 if n != 0 (mod 3), otherwise a(n) = 1 + a(n/3). - Reinhard Zumkeller, Aug 12 2001, edited by M. F. Hasler, Aug 11 2015
From Ralf Stephan, Apr 12 2002: (Start)
a(n) = A051064(n) - 1.
G.f.: Sum_{k>=1} x^3^k/(1 - x^3^k). (End)
Fixed point of the morphism: 0 -> 001; 1 -> 002; 2 -> 003; 3 -> 004; 4 -> 005; etc.; starting from a(1) = 0. - Philippe Deléham, Mar 29 2004
a(n) mod 2 = 1 - A014578(n). - Reinhard Zumkeller, Oct 04 2008
Totally additive with a(p) = 1 if p = 3, 0 otherwise.
v_{m}(n) = Sum_{r>=1} (r/m^(r+1)) Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} exp((2*k*Pi*i*(n+(m-j)*m^r)) / m^(r+1)). This formula is for the general case; for this specific one, set m=3. - A. Neves, Oct 04 2010
a(3n) = A051064(n), a(2n) = a(n), a(2n-1) = A253786(n). - Cyril Damamme, Aug 04 2015
a(3n) = a(n) + 1, a(pn) = a(n) for any other prime p != 3. - M. F. Hasler, Aug 11 2015
3^a(n) = A038500(n). - Antti Karttunen, Oct 09 2017
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Jul 11 2020
a(n) = tau(n)/(tau(3*n) - tau(n)) - 1, where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
a(n) = 3*Sum_{j=1..floor(log_3(n))} frac(binomial(n,3^j)*3^(j-1)/n). - Dario T. de Castro, Jul 10 2022
a(n) = A080342(gcd(n, 3^A080342(n))). - Alan Michael Gómez Calderón, Jul 28 2024

A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 19

Offset: 0

Warut Roonguthai

Also the highest power of 10 dividing n! (different from A054899). - Hieronymus Fischer, Jun 18 2007
Alternatively, a(n) equals the expansion of the base-5 representation A007091(n) of n (i.e., where successive positions from right to left stand for 5^n or A000351(n)) under a scale of notation whose successive positions from right to left stand for (5^n - 1)/4 or A003463(n); for instance, n = 7392 has base-5 expression 2*5^5 + 1*5^4 + 4*5^3 + 0*5^2 + 3*5^1 + 2*5^0, so that a(7392) = 2*781 + 1*156 + 4*31 + 0*6 + 3*1 + 2*0 = 1845. - Lekraj Beedassy, Nov 03 2010
Partial sums of A112765. - Hieronymus Fischer, Jun 06 2012
Also the number of trailing zeros in A000165(n) = (2*n)!!. - Stefano Spezia, Aug 18 2024

			a(100)  = 24.
a(10^3) = 249.
a(10^4) = 2499.
a(10^5) = 24999.
a(10^6) = 249998.
a(10^7) = 2499999.
a(10^8) = 24999999.
a(10^9) = 249999998.
a(10^n) = 10^n/4 - 3 for 10 <= n <= 15 except for a(10^14) = 10^14/4 - 2. - _M. F. Hasler_, Dec 27 2019

M. Gardner, "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, 1978, pp. 50-65.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
David S. Hart, James E. Marengo, Darren A. Narayan, and David S. Ross, On the number of trailing zeros in n!, College Math. J., 39(2):139-145, 2008.
Enrique Pérez Herrero, Trailing Zeros in n!, Psychedelic Geometry Blogspot.
Soichi Ikeda and Kaneaki Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin., 8 (16) 2013, 63-69.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Shu-Chung Liu and Jean C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), 10.5.4, eq (5).
Antonio M. Oller-Marcén, A new look at the trailing zeros of n!, arXiv:0906.4868v1 [math.NT], 2009.
Antonio M. Oller-Marcén and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8
Eric Weisstein's World of Mathematics, Factorial.
Index entries for sequences related to factorial numbers

See A000966 for the missing numbers. See A011371 and A054861 for analogs involving powers of 2 and 3.
Cf. A054899, A007953, A112765, A067080, A098844, A132027, A067080, A098844, A132029, A054999, A112765, A191610, A000351.
Cf. also A000142, A004154.
Cf. A000165, A008904.

a027868 n = sum $ takeWhile (> 0) $ map (n `div`) $ tail a000351_list
-- Reinhard Zumkeller, Oct 31 2012

[Valuation(Factorial(n), 5): n in [0..80]]; // Bruno Berselli, Oct 11 2021

0, seq(add(floor(n/5^i),i=1..floor(log[5](n))), n=1..100); # Robert Israel, Nov 13 2014

Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
Table[ IntegerExponent[n!], {n, 0, 80}] (* Robert G. Wilson v *)
zOF[n_Integer?Positive]:=Module[{maxpow=0},While[5^maxpow<=n,maxpow++];Plus@@Table[Quotient[n,5^i],{i,maxpow-1}]]; Attributes[zOF]={Listable}; Join[{0},zOF[ Range[100]]] (* Harvey P. Dale, Apr 11 2022 *)

a(n)={my(s);while(n\=5,s+=n);s} \\ Charles R Greathouse IV, Nov 08 2012, edited by M. F. Hasler, Dec 27 2019

a(n)=valuation(n!,5) \\ Charles R Greathouse IV, Nov 08 2012

apply( A027868(n)=(n-sumdigits(n,5))\4, [0..99]) \\ M. F. Hasler, Dec 27 2019

from sympy import multiplicity
A027868, p5 = [0,0,0,0,0], 0
for n in range(5,10**3,5):
    p5 += multiplicity(5,n)
    A027868.extend([p5]*5) # Chai Wah Wu, Sep 05 2014

def A027868(n): return 0 if n<5 else n//5 + A027868(n//5) # David Radcliffe, Jun 26 2016

from sympy.ntheory.factor_ import digits
def A027868(n): return n-sum(digits(n,5)[1:])>>2 # Chai Wah Wu, Oct 18 2024

a(n) = Sum_{i>=1} floor(n/5^i).
a(n) = (n - A053824(n))/4.
From Hieronymus Fischer, Jun 25 2007 and Aug 13 2007, edited by M. F. Hasler, Dec 27 2019: (Start)
G.f.: g(x) = Sum_{k>0} x^(5^k)/(1-x^(5^k))/(1-x).
a(n) = Sum_{k=5..n} Sum_{j|k, j>=5} (floor(log_5(j)) - floor(log_5(j-1))).
G.f.: g(x) = L[b(k)](x)/(1-x) where L[b(k)](x) = Sum_{k>=0} b(k)*x^k/(1-x^k) is a Lambert series with b(k) = 1, if k>1 is a power of 5, else b(k) = 0.
G.f.: g(x) = Sum_{k>0} c(k)*x^k/(1-x), where c(k) = Sum_{j>1, j|k} floor(log_5(j)) - floor(log_5(j - 1)).
Recurrence:
a(n) = floor(n/5) + a(floor(n/5));
a(5*n) = n + a(n);
a(n*5^m) = n*(5^m-1)/4 + a(n).
a(k*5^m) = k*(5^m-1)/4, for 0 <= k < 5, m >= 0.
Asymptotic behavior:
a(n) = n/4 + O(log(n)),
a(n+1) - a(n) = O(log(n)), which follows from the inequalities below.
a(n) <= (n-1)/4; equality holds for powers of 5.
a(n) >= n/4 - 1 - floor(log_5(n)); equality holds for n = 5^m-1, m > 0.
lim inf (n/4 - a(n)) = 1/4, for n -> oo.
lim sup (n/4 - log_5(n) - a(n)) = 0, for n -> oo.
lim sup (a(n+1) - a(n) - log_5(n)) = 0, for n -> oo.
(End)
a(n) <= A027869(n). - Reinhard Zumkeller, Jan 27 2008
10^a(n) = A000142(n) / A004154(n). - Reinhard Zumkeller, Nov 24 2012
a(n) = Sum_{k=1..floor(n/2)} floor(log_5(n/k)). - Ammar Khatab, Feb 01 2025

Examples added by Hieronymus Fischer, Jun 06 2012

A345531 Smallest prime power greater than the n-th prime.

Original entry on oeis.org

3, 4, 7, 8, 13, 16, 19, 23, 25, 31, 32, 41, 43, 47, 49, 59, 61, 64, 71, 73, 79, 81, 89, 97, 101, 103, 107, 109, 113, 121, 128, 137, 139, 149, 151, 157, 163, 167, 169, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 256, 263, 269, 271, 277

Offset: 1

Dario T. de Castro, Jun 20 2021

nonn

Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.

The prime-power a(n) is at most the next prime, so this sequence is strictly increasing. See also A366833. - Gus Wiseman, Nov 06 2024

			a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.

Cf. A000040, A007814, A007949, A008864, A343249, A343250, A112765, A343251, A214411, A343252, A343253.
Starting with n instead of prime(n): A000015, A031218, A377468, A377780, A377782.
Opposite (greatest prime-power less than): A065514, A377289, A377781.
For squarefree instead of prime-power: A112926, opposite A112925.
The difference from prime(n) is A377281.
The prime terms have indices A377286(n) - 1.
First differences are A377703.
A version for perfect-powers is A378249.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 and A361102 list the non-prime-powers, differences A375735.
Cf. A366833, A053607, A080101, A304521, A366833, A377057, A377283, A377287, A377288, A377432.

f:= proc(n) local p,x;
  p:= ithprime(n);
  for x from p+1 do
    if nops(numtheory:-factorset(x)) = 1 then return x fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 25 2024

a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
tabseq = Table[a[i],{i, 1, 100}];
(* second program *)
Table[NestWhile[#+1&,Prime[n]+1, Not@*PrimePowerQ],{n,100}] (* Gus Wiseman, Nov 06 2024 *)

A000015(n) = for(k=n,oo,if((1==k)||isprimepower(k),return(k)));
A345531(n) = A000015(1+prime(n)); \\ Antti Karttunen, Jul 19 2021

from itertools import count
from sympy import prime, factorint
def A345531(n): return next(filter(lambda m:len(factorint(m))<=1, count(prime(n)+1))) # Chai Wah Wu, Oct 25 2024

a(n) = A000015(1+A000040(n)). - Antti Karttunen, Jul 19 2021

a(n) = A000015(A008864(n)). - Omar E. Pol, Oct 27 2021

A122840 a(n) is the number of 0's at the end of n when n is written in base 10.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 13 2006

Greatest k such that 10^k divides n.
a(n) = the number of digits in n - A160093(n).
a(A005117(n)) <= 1. - Reinhard Zumkeller, Mar 30 2010
See A054899 for the partial sums. - Hieronymus Fischer, Jun 08 2012
From Amiram Eldar, Mar 10 2021: (Start)
The asymptotic density of the occurrences of k is 9/10^(k+1).
The asymptotic mean of this sequence is 1/9. (End)

			a(160) = 1 because there is 1 zero at the end of 160 when 160 is written in base 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
S. Ikeda and K. Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin., 8 (16) 2013, 63-69.

A007814 is the base 2 equivalent of this sequence.

Cf. A160094, A160093, A001511, A070940, A122841, A027868, A054899, A196563, A196564, A004151, A112765.

a122840 n = if n < 10 then 0 ^ n else 0 ^ d * (a122840 n' + 1)
            where (n', d) = divMod n 10
-- Reinhard Zumkeller, Mar 09 2013

a[n_] := IntegerExponent[n, 10]; Array[a, 100] (* Amiram Eldar, Mar 10 2021 *)

a(n)=valuation(n,10) \\ Charles R Greathouse IV, Feb 26 2014

def a(n): return len(str(n)) - len(str(int(str(n)[::-1]))) # Indranil Ghosh, Jun 09 2017

def A122840(n): return len(s:=str(n))-len(s.rstrip('0')) # Chai Wah Wu, Jul 06 2022

A122840 = lambda n: sympy.multiplicity(10,n) # M. F. Hasler, Apr 05 2024

a(n) = A160094(n) - 1.
From Hieronymus Fischer, Jun 08 2012: (Start)
With m = floor(log_10(n)), frac(x) = x-floor(x):
a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/10^j))).
a(n) = m + Sum_{j=1..m} (floor(-frac(n/10^j))).
a(n) = A054899(n) - A054899(n-1).
G.f.: g(x) = Sum_{j>0} x^10^j/(1-x^10^j). (End)
a(n) = min(A007814(n), A112765(n)). - Jianing Song, Jul 23 2022

A008904 a(n) is the final nonzero digit of n!.

Original entry on oeis.org

1, 1, 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, 2, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6, 4, 2, 2, 4, 2, 8, 4, 4, 8, 4, 6, 6, 6, 2, 6, 4, 6, 6, 2, 6, 4, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6

Offset: 0

Russ Cox

This sequence is not ultimately periodic. This can be deduced from the fact that the sequence can be obtained as a fixed point of a morphism. - Jean-Paul Allouche, Jul 25 2001
The decimal number 0.1126422428... formed from these digits is a transcendental number; see the article by G. Dresden. The Mathematica code uses Dresden's formula for the last nonzero digit of n!; this is more efficient than simply calculating n! and then taking its least-significant digit. - Greg Dresden, Feb 21 2006
From Robert G. Wilson v, Feb 16 2011: (Start)
  (mod 10) == 2          4          6          8
10^
   1         4          2          1          1
   2        28         23         22         25
   3       248        247        260        243
   4      2509       2486       2494       2509
   5     25026      24999      24972      25001
   6    249993     250012     250040     249953
   7   2500003    2499972    2499945    2500078
   8  25000078   24999872   25000045   25000003
   9 249999807  250000018  250000466  249999707 (End)

			6! = 720, so a(6) = 2.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 202.
Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978
S. Kakutani, Ergodic theory of shift transformations, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Univ. Calif. Press, vol. II, 1967, 405-414.
Popular Computing (Calabasas, CA), Problem 120, Factorials, Vol. 4 (No. 36, Mar 1976), page PC36-3.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Washington Bomfim, An algorithm to find the last nonzero digit of n!
Washington Bomfim, A property of the last non-zero digit of factorials
K. S. Brown, The least significant nonzero digit of n!
F. M. Dekking, Regularity and irregularity of sequences generated by automata, Sém. Théor. Nombres, Bordeaux, Exposé 9, 1979-1980, pages 9-01 to 9-10.
Jean-Marc Deshouillers, A footnote to the least non zero digit of n! in base 12, Uniform Distribution Theory 7:1 (2012), pp. 71-73.
Jean-Marc Deshouillers, Yet Another Footnote to the Least Non Zero Digit of n! in Base 12. Unif. Distrib. Theory 11 (2016), no. 2, 163-167.
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Mathematics Magazine, pp. 96-105, vol. 81, 2008.
Gregory P. Dresden, Two Irrational Numbers from the Last Nonzero Digits of n! and n^n, Mathematics Magazine, Vol. 74, No. 4 (2001), 316-320.
Gregory P. Dresden, Research Papers.
Fritz Jacob (fritzjacob(AT)gmail.com), A way to compute a(n)
MathPages, Least Significant Non-Zero Digit of n!
J. C. Martin, The structure of generalized Morse minimal sets on m symbols, Trans. Amer. Math. Soc. 232 (1977), 343-355.
T. Sillke, What are the next entries for the following sequences? Puzzle U asks for the next number after 2642242888682886824484644846.
Eric Weisstein's World of Mathematics, Factorial
David W. Wilson, Minimal state machine for this sequence
David W. Wilson, Another method for computing this sequence
Index entries for sequences related to final digits of numbers
Index entries for sequences related to factorial numbers

Cf. A008905, A000142, A004154, A034886.
Indices of 2,4,6,8: A045547, A045548, A045549, A045550.
Other bases: A136690, A136691, A136692, A136693, A136694, A136695, A136696, A136697, A136698, A136699, A136700, A136701, A136702.

a008904 n = a008904_list !! n
a008904_list = 1 : 1 : f 2 1 where
   f n x = x' `mod` 10 : f (n+1) x' where
      x' = g (n * x) where
         g m | m `mod` 5 > 0 = m
             | otherwise     = g (m `div` 10)
-- Reinhard Zumkeller, Apr 08 2011

f[n_]:=Module[{m=n!},While[Mod[m,10]==0,m=m/10];Mod[m,10]]
Table[f[i],{i,0,100}]
f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; Join[{1, 1}, Table[f[n], {n, 2, 100}]] (* program contributed by Jacob A. Siehler, Greg Dresden, Feb 21 2006 *)
zOF[n_Integer?Positive] := Module[{maxpow=0}, While[5^maxpow<=n,maxpow++]; Plus@@Table[Quotient[n,5^i], {i,maxpow-1}]]; Flatten[Table[ Take[ IntegerDigits[ n!], {-zOF[n]-1}],{n,100}]] (* Harvey P. Dale, Dec 16 2010 *)
f[n_]:=Block[{id=IntegerDigits[n!, 10]}, While[id[[-1]]==0, id=Most@id]; id[[-1]]]; Table[f@n, {n, 0, 100}] (* Vincenzo Librandi, Sep 07 2017 *)

a(n) = r=1; while(n>0, r *= Mod(4, 10)^((n\10)%2) * [1, 2, 6, 4, 2, 2, 4, 2, 8][max(n%10, 1)]; n\=5); lift(r) \\ Charles R Greathouse IV, Nov 05 2010; cleaned up by Max Alekseyev, Jan 28 2012

def a(n):
    if n <= 1: return 1
    return 6*[1,1,2,6,4,4,4,8,4,6][n%10]*3**(n/5%4)*a(n/5)%10
# Maciej Ireneusz Wilczynski, Aug 23 2010

from functools import reduce
from sympy.ntheory.factor_ import digits
def A008904(n): return reduce(lambda x,y:x*y%10,(((6,2,4,8,6,2,4,8,2,4,8,6,6,2,4,8,4,8,6,2)[(a<<2)|(i*a&3)] if i*a else (1,1,2,6,4)[a]) for i, a in enumerate(digits(n,5)[-1:0:-1])),6) if n>1 else 1 # Chai Wah Wu, Dec 07 2023

def A008904(n):
    # algorithm from David Wilson, http://oeis.org/A008904/a008904b.txt
    if n == 0 or n == 1: return 1
    dd = n.digits(base=5)
    x = sum(i*d for i,d in enumerate(dd))
    y = sum(d for d in dd if d % 2 == 0)/2
    z = 2**((x+y) % 4)
    if z == 1: z = 6
    return z # D. S. McNeil, Dec 09 2010

The generating function for n>1 is as follows: for n = a_0 + 5*a_1 + 5^2*a_2 + ... + 5^N*a_N (the expansion of n in base-5), then the last nonzero digit of n!, for n>1, is 6*Product_{i=0..N} (a_i)! (2^(i a_i)) mod 10. - Greg Dresden, Feb 21 2006
a(n) = f(n,1,0) with f(n,x,e) = if n < 2 then A010879(x*A000079(e)) else f(n-1, A010879(x)*A132740(n), e+A007814(n)-A112765(n)). - Reinhard Zumkeller, Aug 16 2008
From Washington Bomfim, Jan 09 2011: (Start)
a(0) = 1, a(1) = 1, if n >= 2, with
n represented in base 5 as (a_h, ..., a_1, a_0)_5,
t = Sum_{i = h, h-1, ... , 0} (a_i even),
x = Sum_{i=h, h-1, ... , 1} (Sum_{k=h, h-1, ..., i}(a_i)),
z = (x + t/2) mod 4, and y = 2^z,
a(n) = 6*(y mod 2) + y*(1-(y mod 2)).
For n >= 5, and n mod 5 = 0,
     i) a(n) = a(n+1) = a(n+3),
    ii) a(n+2) = 2*a(n) mod 10, and
   iii) a(n+4) = 4*a(n) mod 10.
For k not equal to 1, a(10^k) = a(2^k). See second Dresden link, and second Bomfim link.
(End)

More terms from Greg Dresden, Feb 21 2006

cp's OEIS Frontend

A381833 k/25 is in this list if k > 5 and A053824(k) = A112765(k), i.e. if digitsum(k, 5) = valuation(k, 5).

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A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

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A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n.

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A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.

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A345531 Smallest prime power greater than the n-th prime.

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