A004151 -id:A004151 - cp's OEIS frontend

A061816 Obtain m by omitting trailing zeros from n (cf. A004151); a(n) = smallest multiple k*m which is a palindrome.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 252, 494, 252, 525, 272, 272, 252, 171, 2, 252, 22, 161, 696, 525, 494, 999, 252, 232, 3, 434, 2112, 33, 272, 525, 252, 111, 494, 585, 4, 656, 252, 989, 44, 585, 414, 141, 2112, 343, 5, 969, 676, 212, 27972, 55, 616, 171, 232

Offset: 0

Klaus Brockhaus, Jun 25 2001

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see links).

Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29 2001

			For n = 30 we have m = 3, 1*m = 3 is a palindrome, so a(30) = 3. For n = m = 12 the smallest palindromic multiple is 21*m = 252, so a(12) = 252.

P. De Geest, Smallest multipliers to make a number palindromic.

Cf. A050782, A062293, A061915, A061916. Values of k are given in A061906.

stop := 200000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(mp," "); else write(-1," "); end; end;

A083960 Smallest palindromic multiple of (n with trailing 0's omitted, A004151) using only nonzero digits of n; all digits must appear; or 0 if no such number exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 2112, 131131, 414414, 5115, 61616, 7111117, 8118, 99199, 2, 1121211, 22, 32223, 4224, 525, 262262, 22222722222, 828828, 922229, 3, 133331, 23232, 33, 4333334, 535535, 6336, 333777333, 88388, 393393, 4

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 20 2003

If one generates all palindromes using all the digits of n with k digits where k starts with the number of digits of n and increases k until one finds a multiple of n, one finds such palindromic multiples much faster than looking for palindromes among the multiples of n. - W. Edwin Clark

Cf. A061816, A083961, A004151.

More terms from W. Edwin Clark, May 24 2003

Corrected by David Wasserman, Dec 06 2004

A083961 a(n) = A083960(n)/A004151(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 176, 10087, 29601, 341, 3851, 418301, 451, 5221, 2, 53391, 1, 1401, 176, 21, 10087, 823063786, 29601, 31801, 3, 4301, 726, 1, 127451, 15301, 176, 9021009, 2326, 10087, 4, 279051571, 53391, 801, 1, 121, 1401, 952, 176, 203051

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 20 2003

Cf. A083960.

Corrected and extended by David Wasserman, Dec 06 2004

A004719 Delete all 0's from n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 4, 41, 42, 43, 44, 45, 46, 47, 48, 49, 5, 51, 52, 53, 54, 55, 56, 57, 58, 59, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 7, 71, 72, 73, 74, 75, 76, 77, 78, 79, 8, 81, 82, 83, 84, 85, 86, 87, 88, 89, 9, 91, 92, 93, 94, 95, 96, 97, 98, 99, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 111, 112, 113, 114, 115, 116, 117, 118, 119, 12

Offset: 1

N. J. A. Sloane

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

Cf. A004151.

a004719 = read . filter (/= '0') . show :: Integer -> Integer
-- Reinhard Zumkeller, Feb 02 2012

noz:=proc(n) local a,t1,i,j; a:=0; t1:=convert(n,base,10); for i from 1 to nops(t1) do j:=t1[nops(t1)+1-i]; if j <> 0 then a := 10*a+j; fi; od: a; end;
[seq(f(n),n=1..200)]; # N. J. A. Sloane, Jun 11 2014

Table[FromDigits[DeleteCases[IntegerDigits[n], 0]], {n, 120}] (* Alonso del Arte, Nov 10 2018 *)

a(n, base=10) = fromdigits(select(sign, digits(n, base)), base) \\ Rémy Sigrist, Nov 10 2018

def A004719(n): return int(str(n).replace('0','')) # Chai Wah Wu, Feb 20 2024

a(n) = if n <= 9 then n else (if n mod 10 = 0 then a(n/10) else 10*a(floor(n/10)) + n mod 10). [Reinhard Zumkeller, Feb 02 2012]

A038502 Remove 3's from n.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 19, 20, 7, 22, 23, 8, 25, 26, 1, 28, 29, 10, 31, 32, 11, 34, 35, 4, 37, 38, 13, 40, 41, 14, 43, 44, 5, 46, 47, 16, 49, 50, 17, 52, 53, 2, 55, 56, 19, 58, 59, 20, 61, 62, 7, 64, 65, 22, 67, 68, 23, 70, 71, 8, 73, 74, 25, 76

Offset: 1

N. J. A. Sloane

As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

The largest divisor of n not divisible by 3. - Amiram Eldar, Sep 15 2020

			From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (2*3)*G(x^3) - (2*9)*G(x^9) - (2*27)*G(x^27) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (2/3)*H(x^3) - (2/9)*H(x^9) - (2/27)*H(x^27) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (2/3^2)*L(x^3) - (2/9^2)*L(x^9) - (2/27^2)*L(x^27) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} 1/a(n)*x^n = L(x) + (2/3)*L(x^3) + (2/3)*L(x^9) + (2/3)*L(x^27) + ... .
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Peter Bala, A note on the sequence of numerators of a rational function .

Cf. A007949, A038500, A065330.

Result of iterative removal of other factors: A000265 (2), A065883 (4), A132739 (5), A244414 (6), A242603 (7), A004151 (10).

a038502 n = if m > 0 then n else a038502 n'  where (n', m) = divMod n 3
-- Reinhard Zumkeller, Jan 03 2011

[n/3^Valuation(n,3): n in [1..80]]; // Bruno Berselli, May 21 2013

f[n_] := Times @@ (First@#^Last@# & /@ Select[ FactorInteger@n, First@# != 3 &]); Array[f, 76] (* Robert G. Wilson v, Jul 31 2006 *)
Table[n/3^IntegerExponent[n, 3], {n, 100}] (* Amiram Eldar, Sep 15 2020 *)

a(n)=if(n<1, 0, n/3^valuation(n,3)) /* Michael Somos, Nov 10 2005 */

Multiplicative with a(p^e) = 1 if p = 3, otherwise p^e. - Mitch Harris, Apr 19 2005
a(0) = 0, a(3*n) = a(n), a(3*n+1) = 3*n+1, a(3*n+2) = 3*n+2.
Dirichlet g.f. zeta(s-1)*(3^s-3)/(3^s-1). - R. J. Mathar, Feb 11 2011
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,3^n).
O.g.f.: F(x) - 2*F(x^3) - 2*F(x^9) - 2*F(x^27) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,
Sum_{n >= 0} a(n)^m*x^n = F(m,x) - (3^m - 1)( F(m,x^3) + F(m,x^9) + F(m,x^27) + ... ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences n^m*a(n), m in Z. Some examples are given below. (End)
Sum_{k=1..n} a(k) ~ (3/8) * n^2. - Amiram Eldar, Oct 29 2022
a(n) = n / A038500(n). - R. J. Mathar, Mar 13 2024

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

A004185 Arrange digits of n in increasing order, then (for n > 0) omit the zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 12, 22, 23, 24, 25, 26, 27, 28, 29, 3, 13, 23, 33, 34, 35, 36, 37, 38, 39, 4, 14, 24, 34, 44, 45, 46, 47, 48, 49, 5, 15, 25, 35, 45, 55, 56, 57, 58, 59, 6, 16, 26, 36, 46, 56, 66, 67, 68, 69, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Record values: A009994. - Reinhard Zumkeller, Dec 05 2009
If we define "sortable primes" as prime numbers that remain prime when their digits are sorted in increasing order, then all absolute primes (A003459) are sortable primes but not all sortable primes are absolute primes. For example, 311 is both sortable and absolute, and 271 is sortable but not absolute, since its digits can be permuted to 217 = 7 * 31 or 712 = 2^3 * 89, etc. - Alonso del Arte, Oct 05 2013
The above mentioned "sortable primes" are listed in A211654, the nontrivial ones (with digits not in nondecreasing order) in A086042. - M. F. Hasler, Jul 30 2019

			a(19) = 19 because the digits are already in increasing order.
a(20) = 2 because the digits of 20 are 2 and 0, which in increasing order are 0 and 2, but since zero-padding is not allowed on the left, the zero digit is dropped and we are left with 2.
a(21) = 12 because the digits of 21 are 2 and 1, which in increasing order are 1 and 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004719, A004151, A004086, A004186, A009996, A221714, A073138, A193581, A193582, A065641.

Cf. A211654 (sortable primes) and subsequence A086042 (nontrivial solutions).

import Data.List (sort)
a004185 n = read $ sort $ show n :: Integer
-- Reinhard Zumkeller, Aug 10 2011

A004185:=func; [n eq 0 select 0 else A004185(n): n in [0..57]]; // Bruno Berselli, Apr 03 2012

A004185 := proc(n)
    local dgs;
    convert(n,base,10) ;
    dgs := sort(%,`>`) ;
    add( op(i,dgs)*10^(i-1),i=1..nops(dgs)) ;
end proc:
seq(A004185(n),n=0..20) ; # R. J. Mathar, Jul 26 2015

FromDigits[Sort[DeleteCases[IntegerDigits[#], 0]]]&/@Range[0, 60] (* Harvey P. Dale, Nov 29 2011 *)

a(n)=fromdigits(vecsort(digits(n))) \\ Charles R Greathouse IV, Feb 06 2017

def A004185(n):
    return int(''.join(sorted(str(n))).replace('0','')) if n > 0 else 0 # Chai Wah Wu, Nov 10 2015

A160093 Number of digits in n, excluding any trailing zeros.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3

Offset: 1

Anonymous, May 01 2009

			a(1060000) = 3 because discarding the trailing zeros from 1060000 leaves 106, which is a 3-digit number.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A004151, A054899, A055640, A055641, A055642, A102669, A122840, A122841, A160094, A196563, A196564.

lnzd[n_]:=Module[{spl=Last[Split[IntegerDigits[n]]]},If[!MemberQ[ spl,0], IntegerLength[n], IntegerLength[n]-Length[spl]]]; Array[lnzd,110] (* Harvey P. Dale, Jun 05 2013 *)
Table[IntegerLength[n] - IntegerExponent[n, 10], {n, 100}] (* Amiram Eldar, Sep 14 2020 *)

a(n)=if(n==0,1,#digits(n/10^valuation(n,10))) \\ Joerg Arndt, Jan 11 2017

a(n)=logint(n,10)+1-valuation(n,10) \\ Charles R Greathouse IV, Jan 12 2017

def A160093(n):
     return len(str(int(str(n)[::-1]))) # Indranil Ghosh, Jan 11 2017

From Hieronymus Fischer, Jun 08 2012: (Start)
With m = floor(log_10(n)), frac(x) = x-floor(x):
a(n) = 1 + Sum_{j=0..m} ceiling(frac(n/10^j)).
a(n) = 1 - Sum_{j=1..m} (floor(-frac(n/10^j))).
a(n)= A055642(n) + A054899(n-1) - A054899(n).
G.f.: (x/(1-x)) + (1/(1-x))*Sum_{j>0} x^(10^j+1)*(1 - x^(10^j-1))/(1-x^10^j). (End)
a(n) = A055642(A004086(n)). - Indranil Ghosh, Jan 11 2017
a(n) = A055642(A004151(n)). - Amiram Eldar, Sep 14 2020

Simpler definition and changed example from Jon E. Schoenfield, Feb 15 2014

A004154 a(n) = n! with trailing zeros omitted.

Original entry on oeis.org

1, 1, 2, 6, 24, 12, 72, 504, 4032, 36288, 36288, 399168, 4790016, 62270208, 871782912, 1307674368, 20922789888, 355687428096, 6402373705728, 121645100408832, 243290200817664, 5109094217170944, 112400072777760768, 2585201673888497664, 62044840173323943936

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Index entries for sequences related to factorial numbers

Cf. A000142, A004151, A008904 (mod 10).

a004154 = a004151 . a000142
a004154_list = scanl (\u v -> a004151 $ u * v) 1 [1..]
-- Reinhard Zumkeller, Nov 24 2012

[Factorial(n)/10^Valuation(Factorial(n), 5): n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

a:= n-> (f-> f/10^padic[ordp](f,10))(n!):
seq(a(n), n=0..29);  # Alois P. Heinz, Dec 29 2021

Array[#!//.x_/;x~Mod~5==0:>x/10&,22]  (* Giorgos Kalogeropoulos, Aug 17 2020 *)
Join[{1,1,2,6,24},Table[FromDigits[Flatten[Most[Split[IntegerDigits[n!]]]]],{n,5,30}]] (* or *) Table[n!/10^IntegerExponent[n!,10],{n,0,30}] (* Harvey P. Dale, Feb 13 2024 *)

a(n)=n!/10^valuation(n!,5) \\ M. F. Hasler, Oct 16 2014

from sympy import factorial
from sympy.ntheory.factor_ import digits
def A004154(n): return factorial(n)//10**(n-sum(digits(n,5)[1:])>>2) # Chai Wah Wu, Oct 18 2024

from itertools import count, islice
def agen(): # generator of terms
    f = 1
    for n in count(1):
        yield f
        while n%10 == 0: n //= 10
        f *= n
        while f%10 == 0: f //= 10
print(list(islice(agen(), 25))) # Michael S. Branicky, Apr 11 2025

a(n) = A000142(n) / 10^A027868(n). - Reinhard Zumkeller, Nov 24 2012
a(n+1) = A004151((n+1)*a(n)). - Reinhard Zumkeller, Nov 24 2012, corrected by M. F. Hasler, Oct 16 2014
a(n) = A004151(A000142(n)) = A000142(n)/A011557(A112765(n)), or A122840 instead of A112765. - M. F. Hasler, Oct 16 2014

A061917 Either a palindrome or becomes a palindrome if trailing 0's are omitted.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, 363, 373, 383, 393, 400, 404

Offset: 1

N. J. A. Sloane, Jun 27 2001

Numbers that are palindromes when written with a suitable number of leading zeros. - Jeppe Stig Nielsen, Jan 17 2022

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

Cf. A002113, A057890, A057891.

a061917 n = a061917_list !! (n-1)
a061917_list = filter chi [0..] where
   chi x = zs == reverse zs where
      zs = dropWhile (== '0') $ reverse $ show x
-- Reinhard Zumkeller, Sep 25 2011

PaleQ[n_Integer, base_Integer] := Module[{idn, trim = n/base^IntegerExponent[n, base]}, idn = IntegerDigits[trim, base]; idn == Reverse[idn]]; Select[Range[0, 500], PaleQ[#, 10] &] (* Lei Zhou, Dec 13 2013 *)
Join[{0},Select[Range[500],PalindromeQ[FromDigits[Drop[IntegerDigits[#],-IntegerExponent[#,10]]]]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2017 *)

isOK(k)=k==0||fromdigits(Vecrev(digits(k)))==k/10^valuation(k,10) \\ Jeppe Stig Nielsen, Jan 17 2022

def ispal(s): return s == s[::-1]
def ok(n): s = str(n); return ispal(s) or ispal(s.rstrip('0'))
print([k for k in range(405) if ok(k)]) # Michael S. Branicky, Jan 17 2022

A136522(A004151(a(n))) = 1. - Reinhard Zumkeller, Sep 25 2011

Corrected by Ray Chandler, Jun 08 2009

cp's OEIS Frontend

A061816 Obtain m by omitting trailing zeros from n (cf. A004151); a(n) = smallest multiple k*m which is a palindrome.

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A083960 Smallest palindromic multiple of (n with trailing 0's omitted, A004151) using only nonzero digits of n; all digits must appear; or 0 if no such number exists.

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A083961 a(n) = A083960(n)/A004151(n).

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A004719 Delete all 0's from n.

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A038502 Remove 3's from n.

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

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A004185 Arrange digits of n in increasing order, then (for n > 0) omit the zeros.

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