A010879 -id:A010879 - cp's OEIS frontend

A051624 12-gonal (or dodecagonal) numbers: a(n) = n(5n-4).

0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, 4089, 4380, 4681, 4992, 5313, 5644, 5985, 6336, 6697, 7068, 7449, 7840, 8241, 8652

Offset: 0

Barry E. Williams

Zero followed by partial sums of A017281. - Klaus Brockhaus, Nov 20 2008
Sequence found by reading the line from 0, in the direction 0, 12, ... and the parallel line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - Omar E. Pol, Jul 18 2012
This is also a star hexagonal number: a(n) = A000384(n) + 6*A000217(n-1). - Luciano Ancora, Mar 30 2015
Starting with offset 1, this is the binomial transform of (1, 11, 10, 0, 0, 0, ...). - Gary W. Adamson, Aug 01 2015
a(n+1) is the sum of the odd numbers from 4n+1 to 6n+1. - Wesley Ivan Hurt, Dec 14 2015
For n >= 2, a(n) is the number of intersection points of all unit circles centered on the inner lattice points of an (n+1) X (n+1) square grid. - Wesley Ivan Hurt, Dec 08 2020
The final digit of a(n) equals the final digit of n, A010879(n). - Enrique Pérez Herrero, Nov 13 2022
a(n-1) is the maximum second Zagreb index of maximal 2-degenerate graphs with n vertices.  (The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.) - Allan Bickle, Apr 16 2024

			The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

T. D. Noe, Table of n, a(n) for n = 0..1000
John Elias, Illustration: compass configuration , Illustration: cross configuration.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Wikipedia, Dodecagonal number
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A007587.
Cf. A093645 ((10, 1) Pascal, column m=2). Partial sums of A017281.
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
Cf. A372027 (second Zagreb index of MOPs).

[ n eq 1 select 0 else Self(n-1)+10*(n-2)+1: n in [1..43] ]; // Klaus Brockhaus, Nov 20 2008

RecurrenceTable[{a[0]==0, a[1]==1, a[2]==12, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3]}, a, {n, 30}] (* G. C. Greubel, Jul 31 2015 *)
Table[n*(5*n - 4), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

a(n)=(5*n-4)*n \\ Charles R Greathouse IV, Jun 16 2011

G.f.: x*(1+9*x)/(1-x)^3.
a(n) = Sum_{k=0..n-1} 10*k+1. - Klaus Brockhaus, Nov 20 2008
a(n) = 10*n + a(n-1) - 9 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(n) = A131242(10n). - Philippe Deléham, Mar 27 2013
a(10*a(n) + 46*n + 1) = a(10*a(n) + 46*n) + a(10*n+1). - Vladimir Shevelev, Jan 24 2014
E.g.f.: x*(5*x + 1) * exp(x). - G. C. Greubel, Jul 31 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=12. - G. C. Greubel, Jul 31 2015
Sum_{n>=1} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/8 + 5*log(5)/16 + sqrt(5)*log((1 + sqrt(5))/2)/8 = 1.177956057922663858735173968... . - Vaclav Kotesovec, Apr 27 2016
a(n) + 4*(n-1)^2 = (3*n-2)^2. Let P(k,n) be the n-th k-gonal number. Then, in general, P(4k,n) + (k-1)^2*(n-1)^2 = (k*n-k+1)^2. - Charlie Marion, Feb 04 2020
Product_{n>=2} (1 - 1/a(n)) = 5/6. - Amiram Eldar, Jan 21 2021
a(n) = (3*n-2)^2 - (2*n-2)^2. In general, if we let P(k,n) = the n-th k-gonal number, then P(4k,n) = (k*n-(k-1))^2 - ((k-1)*n-(k-1))^2. - Charlie Marion, Nov 11 2021

A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

The length of n-th row is given in A055642(n). - Reinhard Zumkeller, Jul 04 2012

According to the formula for T(n,1), columns are numbered starting with 1. One might also number columns starting with the offset 0, as to have the coefficient of 10^k in column k. - M. F. Hasler, Jul 21 2013

Reinhard Zumkeller, Rows n = 0..2500 of triangle, flattened

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031087 for the base-2 to base-9 analogs.

a031298 n k = a031298_tabf !! n !! k
a031298_row n = a031298_tabf !! n
a031298_tabf = iterate succ [0] where
   succ []     = [1]
   succ (9:ds) = 0 : succ ds
   succ (d:ds) = (d + 1) : ds
-- Reinhard Zumkeller, Jul 04 2012

Table[Reverse[IntegerDigits[n]],{n,0,50}]//Flatten (* Harvey P. Dale, Mar 07 2023 *)

T(n,k)=n\10^(k-1)%10 \\ M. F. Hasler, Jul 21 2013

T(n,1) = A010879(n); T(n,A055642(n)) = A000030(n). - Reinhard Zumkeller, Jul 04 2012

Initial 0 and better name by Philippe Deléham, Oct 20 2011

Edited by M. F. Hasler, Jul 21 2013

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

A034709 Numbers divisible by their last digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21, 22, 24, 25, 31, 32, 33, 35, 36, 41, 42, 44, 45, 48, 51, 52, 55, 61, 62, 63, 64, 65, 66, 71, 72, 75, 77, 81, 82, 84, 85, 88, 91, 92, 93, 95, 96, 99, 101, 102, 104, 105, 111, 112, 115, 121, 122, 123, 124, 125, 126, 128, 131, 132

Offset: 1

Erich Friedman

Union of A017281, A017293, A139222, A139245, A017329, A139249, A139264, A139279 and A139280. - Reinhard Zumkeller, Jun 22 2008
The differences between consecutive terms repeat with period 1177 and the corresponding terms differ by 2520 = LCM(1,2,...,9). In other words, a(k*1177+i) = 2520*k + a(i). - Giovanni Resta, Aug 20 2015
The asymptotic density of this sequence is 1177/2520 = 0.467063... (see A341431 and A341432 for the values in other base representations). - Amiram Eldar, Nov 24 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010879, A034838, A007602.
Cf. A017281, A017293, A139222, A139245, A017329, A139249, A139264, A139279, A139280.
Cf. A341431, A341432.

import Data.Char (digitToInt)
a034709 n = a034709_list !! (n-1)
a034709_list =
   filter (\i -> i `mod` 10 > 0 && i `mod` (i `mod` 10) == 0) [1..]
-- Reinhard Zumkeller, Jun 19 2011

N:= 1000: # to get all terms <= N
sort([seq(seq(ilcm(10,d)*x+d, x=0..floor((N-d)/ilcm(10,d))), d=1..9)]); # Robert Israel, Aug 20 2015

dldQ[n_]:=Module[{idn=IntegerDigits[n],last1},last1=Last[idn]; last1!= 0&&Divisible[n,last1]]; Select[Range[150],dldQ]  (* Harvey P. Dale, Apr 25 2011 *)
Select[Range[150],Mod[#,10]!=0&&Divisible[#,Mod[#,10]]&] (* Harvey P. Dale, Aug 07 2022 *)

for(n=1,200,if(n%10,if(!(n%digits(n)[#Str(n)]),print1(n,", ")))) \\ Derek Orr, Sep 19 2014

A034709_list = [n for n in range(1, 1000) if n % 10 and not n % (n % 10)]
# Chai Wah Wu, Sep 18 2014

A003893 a(n) = Fibonacci(n) mod 10.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, elipper(AT)uoft02.utoledo.edu

All blocks of 60 successive terms contain 20 even and 40 odd numbers. - Reinhard Zumkeller, Apr 09 2005

These are the analogs of the Fibonacci numbers in carryless arithmetic mod 10.

G. Marsaglia, The mathematics of random number generators, pp. 73-90 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Math. Mag., pp. 96-105, vol. 81, 2008.
Ron Knott, The Mathematical Magic of the Fibonacci Numbers.
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for sequences related to final digits of numbers
Index entries for sequences related to carryless arithmetic
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0,0,1,0,0,-1,0,0,-1,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1).

Cf. A000045, A010879, A089911, A105471, A105472.

Cf. A008963, A261606.

a003893 n = a003893_list !! n
a003893_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 10)
                       (tail a003893_list) a003893_list
-- Reinhard Zumkeller, Jul 01 2013

[Fibonacci(n) mod 10: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

with(combinat,fibonacci); A003893 := proc(n) fibonacci(n) mod 10; end;

Table[Mod[Fibonacci[n], 10], {n, 0, 99}] (* Alonso del Arte, Jul 29 2013 *)
Table[IntegerDigits[Fibonacci[n]][[-1]], {n, 0, 99}] (* Alonso del Arte, Jul 29 2013 *)
NumberDigit[Fibonacci[Range[0,120]],0] (* Requires Mathematica version 12 or later *) (* Harvey P. Dale, Jul 05 2021 *)

a(n)=fibonacci(n)%10 \\ Charles R Greathouse IV, Feb 03 2014

A003893_list, a, b, = [], 0, 1
for _ in range(10**3):
    A003893_list.append(a)
    a, b = b, (a+b) % 10 # Chai Wah Wu, Nov 26 2015

Periodic with period 60 = A001175(10).
From Reinhard Zumkeller, Apr 09 2005: (Start)
a(n) = (a(n-1) + a(n-2)) mod 10 for n > 1, a(0) = 0, a(1) = 1.
a(n) = A105471(n) - A105472(n)*10 = A105471(n)/10. (End)
a(n) = A010879(A000045(n)). - Michel Marcus, Nov 19 2022

More terms from Ray Chandler, Nov 15 2003

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)
All terms must be 1, 2, 4, 6, or 8. - Harvey P. Dale, Sep 05 2025

			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 *)
Mod[#/10^IntegerExponent[#]&/@(Range[0,100]!),10] (* Harvey P. Dale, Sep 05 2025 *)

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

A087097 Lunar primes (formerly called dismal primes) (cf. A087062).

Original entry on oeis.org

19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 209, 219, 309, 319, 329, 409, 419, 429, 439, 509, 519, 529, 539, 549, 609, 619, 629, 639, 649, 659, 709, 719, 729, 739, 749, 759, 769, 809, 819, 829, 839, 849, 859, 869, 879, 901, 902, 903, 904, 905, 906, 907, 908, 909, 912, 913, 914, 915, 916, 917, 918, 919, 923, 924, 925, 926, 927, 928, 929, 934, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 956, 957, 958, 959, 967, 968, 969, 978, 979, 989

Offset: 1

Marc LeBrun, Oct 20 2003

9 is the multiplicative unit. A number is a lunar prime if it is not a lunar product (see A087062 for definition) r*s where neither r nor s is 9.
All lunar primes must contain a 9, so this is a subsequence of A011539.
Also, numbers k such that the lunar sum of the lunar prime divisors of k is k. - N. J. A. Sloane, Aug 23 2010
We have changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing. - N. J. A. Sloane, Aug 06 2014
(Lunar) composite numbers are not necessarily a product of primes. (For example 1 = 1*x for any x in {1, ..., 9} is not a prime but can't be written as the product of primes.) Therefore, to establish primality, it is not sufficient to consider only products of primes; one has to consider possible products of composite numbers as well. - M. F. Hasler, Nov 16 2018

			8 is not prime since 8 = 8*8. 9 is not prime since it is the multiplicative unit. 10 is not prime since 10 = 10*8. Thus 19 is the smallest prime.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..22095 [all primes with at most 5 digits]
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087019, A087061, A087062, A087636, A087638, A087984.

A87097=select( is_A087097(n)={my(d); if( n<100, n>88||(n%10==9&&n>9), vecmax(d=digits(n))<9, 0, #d<5, vecmin(d)A087062(m,k)==n&&return))))}, [1..999]) \\ M. F. Hasler, Nov 16 2018

The set { m in A011539 | 9A054054(m) < min(A000030(m),A010879(m)) } (9ish numbers A011539 with 2 digits or such that the smallest digit is strictly smaller than the first and the last digit) is equal to this sequence up to a(1656) = 10099. The next larger 9ish number 10109 is also in that set but is the lunar square of 109, thus not in this sequence of primes. - M. F. Hasler, Nov 16 2018

A037904 Greatest digit of n - least digit of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(n) = A054055(n)-A054054(n); a(A010785(n)) = 0; for k>0: a(n) = a(n*10^k + A000030(n)) = a(n*10^k + A010879(n)) = a(n*10^k + A054054(n)) = a(n*10^k + A054055(n)) . - Reinhard Zumkeller, Dec 14 2007; corrected by David Wasserman, May 21 2008

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A040163, A064834.

a037904 = f 9 0 where
   f u v 0 = v - u
   f u v z = f (min u d) (max v d) z' where (z', d) = divMod z 10
-- Reinhard Zumkeller, Dec 16 2013

f:= n -> (max-min)(convert(n,base,10)):
map(f, [$1..1000]); # Robert Israel, Jul 07 2016

f[n_] := Block[{d = IntegerDigits[n]}, Max[d] - Min[d]]; Table[ f[n], {n, 1, 15}]

a(n)=my(d=digits(n)); vecmax(d)-vecmin(d) \\ Charles R Greathouse IV, Feb 07 2017

def A037904(n): return int(max(s:=str(n)))-int(min(s)) # Chai Wah Wu, Nov 10 2023

Incorrect comments deleted by Robert Israel, Jul 07 2016

A168046 Characteristic function of zerofree numbers in decimal representation.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Dec 01 2009

a(A052382(n)) = 1; a(A011540(n)) = 0;
a(n) = A000007(A055641(n));
not the same as A168184: a(n)=A168184(n) for n<=100.
a(A007602(n)) = a(A038186(n)) = 1. - Reinhard Zumkeller, Apr 07 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Zerofree
Index entries for characteristic functions

Cf. A052382, A217096, A011540.

a168046 = fromEnum . ch0 where
   ch0 x = x > 0 && (x < 10 || d > 0 && ch0 x') where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015, Apr 07 2011

Map[Boole[FreeQ[IntegerDigits[#], 0]] &, Range[0, 100]] (* Paolo Xausa, May 06 2024 *)

a(n) = A057427(A010879(n)) * (if n<10 then 1 else a(A059995(n))).
From Hieronymus Fischer, Jan 23 2013: (Start)
a(n) = A057427(A007954(n)) = sign(dp_10(n)).
where dp_10(n) digital product of n in base 10.
a(n) = 1 - A217096(n).
a(n) = 1 - sign(A055641(n)).
g(x) = x(1-x^9)/((1-x)(1-x^10))(1 + sum_{j>=1} (x^((10^j-10)/9) - x^10^j)/(1-x^10^(j+1)))).
g(x) = 1/(1-x) - g_A217096(x), where g_A217096(x) is the g.f. of A217096.
(End)

A059995 Drop the final digit of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10

Offset: 0

Henry Bottomley, Mar 12 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A004526, A010879, A054899.

Table[FromDigits[Most[IntegerDigits[n]]],{n,0,110}] (* Harvey P. Dale, Sep 09 2016 *)

a(n)=n\10+(n<0 && n%10) \\ Corrected by M. F. Hasler, May 19 2021

a(n) = sign(n)*(abs(n)\10) \\ David A. Corneth, May 20 2021

def A059995(n): return n//10 # Chai Wah Wu, Jan 20 2023

a(n) = a(n-10) + 1.
a(n) = floor(n/10).
a(n) = (n - A010879(n))/10.
G.f.: x^10/((1-x)(1-x^10)).
Partial sums are given by A131242. - Hieronymus Fischer, Jun 21 2007

cp's OEIS Frontend

A051624 12-gonal (or dodecagonal) numbers: a(n) = n*(5*n-4).

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A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

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

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A034709 Numbers divisible by their last digit.

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A003893 a(n) = Fibonacci(n) mod 10.

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

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A051624 12-gonal (or dodecagonal) numbers: a(n) = n(5n-4).

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