A004719 -id:A004719 - cp's OEIS frontend

A243846 Numbers for which the nozero power-sequence of n falls into a loop.

1, 366784, 14877, 531136, 29287878125, 13631616, 18916327, 1245376, 118971, 1, 24871, 1942272, 377414623, 361123756, 221285675921484375, 453559756, 16185473, 4136832, 113758939, 366784, 164961711, 3179798512, 131147731, 1841716224, 283439365914625, 118754727776

Offset: 1

Anthony Sand, Jun 12 2014

Numbers returned by the following procedure: For n = 1, 2, 3, ..., let x(n; 1) = 1. Begin the recursive sequence x(n; i) = nozero(x(n; i-1) * n), where the function nozero(x) removes all zeros from x (see A004719). When x(n; i) = x(n; j

a(10*n) = a(n). - Pontus von Brömssen, May 19 2019

			a(2) = 366784 because x(2; 491) = nozero(183392 * 2) = 366784. Subsequently x(2; 527) = nozero(1533392 * 2) = nozero(3066784) = 366784, and this happens for the first time. Therefore x(2; 527) = x(2; 491) and the procedure returns x(2; 527) = 366784.
a(3) = 14877 because x(3; 28) = nozero(469359 * 3) = nozero(1408077) = 14877. Subsequently, x(3; 108) = nozero(4959 * 3) = 14877, and this happens for the first time. Therefore x(3; 28) = x(3; 108) and the procedure returns x(3;108) = 14877.

Pontus von Brömssen, Table of n, a(n) for n = 1..128
Sean A. Irvine, Java program (github)

Cf. A004719, A242350, A243845, A306569.

a[n_] := Block[{h = <||>, t = n}, While[! KeyExistsQ[h,t], h[t]=0; t = FromDigits@ Select[ IntegerDigits[n t], # > 0 &]]; t]; Array[a, 20] (* Giovanni Resta, May 20 2019 *)

Recurrence: x(n; i) = nozero(x(n; i-1) * n), x(n; 1) = 1, i >= 2, with n >= 1. For example, for x(2;10) = 512 and nozero(512 * 2) = nozero(1024) = 124. Therefore x(2;11) = 124.

If the sequence {x(n; i)}_{i >= 1} becomes periodic at some entry x(n; j), that is if there exists a period length L(n) such that x(n; i + L(n)) = x(n; i) for i >= j then a(n) = x(n; j). If there is no such period length then one puts a(n) = 0.

Edited: Comment, formula and example reformulated. - Wolfdieter Lang, Jul 13 2014
a(5), a(6), a(8), a(9) corrected by Pontus von Brömssen, May 19 2019
a(10)-a(26) from Giovanni Resta, May 20 2019

A125290 Numbers with at least two distinct digits in decimal representation, none of which is 0.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123

Offset: 1

Reinhard Zumkeller, Nov 26 2006

Also numbers having at least two partitions into digit values of their decimal representations: A061827(a(n)) > 1.

First differs from A101594 at a(83) = 123 != 131 = A101594(83). - Michael S. Branicky, Dec 13 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Subsequence of A052382. Supersequence of A101594.

Cf. A125293, A004719, A043537, A061827, A168046.

a125290 n = a125290_list !! (n-1)
a125290_list = filter ((> 1) . a043537) a052382_list
-- Reinhard Zumkeller, Jun 18 2013

Select[Range[200], FreeQ[#, 0] && Length[Union[#]] > 1 & [IntegerDigits[#]] &] (* Paolo Xausa, May 06 2024 *)

def ok(n): s = set(str(n)); return len(s) >= 2 and "0" not in s
print([k for k in range(124) if ok(k)]) # Michael S. Branicky, Dec 13 2021

A043537(A004719(a(n))) > 1.
A168046(a(n)) * A043537(A004719(a(n))) > 1. - Reinhard Zumkeller, Jun 18 2013
a(n) ~ n. - Charles R Greathouse IV, Feb 13 2017

Name clarified by Michael S. Branicky, Dec 13 2021

A004177 Omit 2's from n.

Original entry on oeis.org

0, 1, 0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 15, 16, 17, 18, 19, 0, 1, 0, 3, 4, 5, 6, 7, 8, 9, 30, 31, 3, 33, 34, 35, 36, 37, 38, 39, 40, 41, 4, 43, 44, 45, 46, 47, 48, 49, 50, 51, 5, 53, 54, 55, 56, 57, 58, 59, 60, 61, 6, 63, 64, 65, 66, 67, 68, 69, 70, 71, 7, 73, 74

Offset: 0

N. J. A. Sloane

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

Cf. A004176, A004178, A004179, A004180, A004181, A004182, A004183, A004184.

Cf. A004719.

f:= proc(n) local L,i;
     L:= subs(2=NULL,convert(n,base,10));
     add(L[i]*10^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Sep 15 2024

Table[FromDigits[DeleteCases[IntegerDigits[n],2]],{n,0,80}] (* Harvey P. Dale, Feb 12 2022 *)

A243845 Numbers generated by recursive procedure a(n) = nozero(a(n-1) * 3), in which the function nozero(x) removes all zeros from x, starting with a(1) = 1.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 5949, 17847, 53541, 16623, 49869, 14967, 4491, 13473, 4419, 13257, 39771, 119313, 357939, 173817, 521451, 1564353, 469359, 14877, 44631, 133893, 41679, 12537, 37611, 112833, 338499, 115497, 346491, 139473, 418419

Offset: 1

Anthony Sand, Jun 12 2014

Numbers in the following sequence: Let a(1) = 1, then begin the recursive sequence a(n) = nozero(a(n-1) * 3), where the function nozero(x) removes all zeros from x.
The sequence returns standard powers of 3 until step 11, where a(11) = nozero(19683 * 3) = nozero(59049) = 5949.
At step 28, a(28) = nozero(469359 * 3) = nozero(1408077) = 14877. At step 108, a(108) = nozero(4959 * 3) = 14877. Therefore a(28) = a(108) and the sequence repeats. Because this is the first instance where a member of this sequence is repeated one has a(n + L) = a(n) for n >= 28 with the primitive (least) period length L = 108 - 28 = 80.

			a(2) = nozero(3*a(1)) = nozero(3) = 3.

Anthony Sand, Table of n, a(n) for n = 1..108
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000244, A004719, A242350, A243846.

NestList[FromDigits@ DeleteCases[IntegerDigits[3 #], ?(# == 0 &)] &, 1, 38] (* _Michael De Vlieger, Jun 27 2020 *)

L=[1]
for i in [1..108]:
    T=(3*L[i-1]).digits(base=10)
    TT=filter(lambda a: a != 0, T)
    L.append(sum(TTi*10^i for i, TTi in enumerate(TT)))
L # - Tom Edgar, Jun 17 2014

a(n) = A004719(a(n-1) * 3) for n>1, a(1) = 1.

Edited: Name, comments and formula reformulated. - Wolfdieter Lang, Jul 13 2014

A247796 From right to left in decimal representation of n: replace each maximal set of adjacent digits with their sum, if this sum is less than 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 4, 5, 6, 7, 8, 9, 46, 47, 48, 49, 5, 6, 7, 8, 9, 55, 56, 57, 58, 59, 6, 7, 8, 9, 64, 65, 66, 67, 68, 69, 7, 8, 9, 73, 74, 75, 76, 77

Offset: 0

Reinhard Zumkeller, Oct 08 2014

			7654321: 7654[321] -> 7654[3+2+1] -> 76546 -> 76[54]6 -> 76[5+4]6 -> 7696 = a(7654321);
1234567: 123[45]67 -> 123[4+5]67 -> 123967 -> [123]967 -> [1+2+3]967 -> 6967 = a(1234567);
1111111: [1111111] -> [1+1+1+1+1+1+1] -> 7 = a(1111111);
a(7777777) = 7777777;
90909: 909[09] -> 909[0+9] -> 9099 -> 9[09]9 -> 9[0+9]9 -> 999 = a(90909);
20202: [20202] -> [2+0+2+0+2] -> 6 = a(20202).

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

Cf. A248013, A248014.

a247796 = f 0 where
   f s 0 = s
   f s x = if s + d < 10 then f (s + d) x' else (f d x') * 10 + s
           where (x', d) = divMod x 10

A247796(n,d=digits(n))={forstep(k=#d,2,-1,if(d[k-1]+d[k]<10, d[k-1]+=d[k]; d=d[^k]));fromdigits(d)} /* or: (about 10% faster) */
A247796(n,u=1)={until(n<10*u*=10,my(m=n\u);while(m>9&&sumdigits(m%100)<10, m=vecsum(divrem(m,10));n=m*u+n%u));n} \\ Trying to reduce the number of redefinitions of n yields slower code. M. F. Hasler, Jan 29 2018

a(n) <= n; a(A248013(n)) = A248013(n); a(A248014(n)) < A248014(n);

a(n) = a(a(n)) = a(A004719(n)) = a(n * 10^k).

Edited by M. F. Hasler, Jan 29 2018

A356757 Omit zero digits from factorial numbers.

Original entry on oeis.org

1, 1, 2, 6, 24, 12, 72, 54, 432, 36288, 36288, 399168, 47916, 622728, 871782912, 137674368, 2922789888, 35568742896, 64237375728, 121645148832, 243292817664, 5199421717944, 11247277776768, 258521673888497664, 624484173323943936, 15511214333985984, 4329146112665635584

Offset: 0

Stefano Spezia, Aug 26 2022

			a(12) = 47916 since 12! = 479001600.

Index entries for sequences related to factorial numbers

Cf. A000142, A004719.
Cf. A004154, A243657, A321475.
Cf. A027869 (number of omitted zero digits), A356758 (number of nonzero digits).

Table[FromDigits[Select[IntegerDigits[n!],Positive]], {n,0,26}]

a(n) = fromdigits(select(x->(x>0), digits(n!))); \\ Michel Marcus, Aug 26 2022

from math import factorial
def a(n): return int(str(factorial(n)).replace("0", ""))
print([a(n) for n in range(27)]) # Michael S. Branicky, Aug 26 2022

a(n) = A004719(A000142(n)).

A373170 Main diagonal of A373169.

Original entry on oeis.org

1, 3, 9, 22, 45, 81, 43, 24, 288, 118, 138, 585, 184, 918, 288, 232, 294, 351, 559, 2586, 1179, 3586, 117, 72, 3913, 2949, 5949, 1585, 7239, 8811, 3595, 3789, 3537, 5758, 1968, 18, 6454, 4152, 4374, 1687, 549, 7794, 3922, 8313, 828, 2674, 4251, 2646, 5548, 3636, 3879, 799

Offset: 2

Paolo Xausa, May 28 2024

a(n) is the zeroless analog of the (n-1)-th n-gonal number.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A064808, A004719, A373169.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A373170[n_] := Fold[noz[#2*(n-2) + 1 + #] &, 1, Range[n-2]];
Array[A373170, 100, 2]

a(n) = A373169(n,n-1).

A275536 Differences of the exponents of the adjacent distinct powers of 2 in the binary representation of n (with -1 subtracted from the least exponent present) are concatenated as decimal digits in reverse order.

Original entry on oeis.org

1, 2, 11, 3, 12, 21, 111, 4, 13, 22, 112, 31, 121, 211, 1111, 5, 14, 23, 113, 32, 122, 212, 1112, 41, 131, 221, 1121, 311, 1211, 2111, 11111, 6, 15, 24, 114, 33, 123, 213, 1113, 42, 132, 222, 1122, 312, 1212, 2112, 11112

Offset: 1

Armands Strazds, Aug 01 2016

A preferable representation is a sequence of arrays, since multi-digit items are possible: [1],[2],[1,1],[3],[1,2],[2,1],[1,1,1],[4],[1,3],[2,2],[1,1,2],[3,1],[1,2,1],[2,1,1],[1,1,1,1],[5],[1,4],[2,3],[1,1,3],[3,2],[1,2,2],[2,1,2],[1,1,1,2],[4,1],[1,3,1],[2,2,1],[1,1,2,1],[3,1,1],[1,2,1,1],[2,1,1,1],[1,1,1,1,1],[6],[1,5],[2,4],[1,1,4],[3,3],[1,2,3],[2,1,3],[1,1,1,3],[4,2],[1,3,2],[2,2,2],[1,1,2,2],[3,1,2],[1,2,1,2],[2,1,1,2],[1,1,1,1,2]. 0 is not allowed as a digit.
a(512) is the first term which cannot be expressed unambiguously in decimal. - Charles R Greathouse IV, Aug 02 2016
The first two terms which are equal (because of the ambiguity inherent in using decimal, or more generally any finite base) are a(3) = a(1024) = 11. a(3) corresponds to the array [1,1] while a(1024) corresponds to [11]. - Charles R Greathouse IV, Mar 19 2017

			5 = 2^2 + 2^0, so the representation is [2-0, 0-(-1)] = [2, 1] so a(5) = 12.
6 = 2^2 + 2^1, so the representation is [2-1, 1-(-1)] = [1, 2] so a(6) = 21.
18 = 2^4 + 2^1, so the representation is [4-1, 1-(-1)] = [3, 2] so a(18) = 23.

Charles R Greathouse IV, Table of n, a(n) for n = 1..511

Cf. A069800, A261300, A248646, A256494, A258055, A004086, A004719, A071160.

a(n)=my(v=List(),k); while(n, k=valuation(n,2)+1; n>>=k; listput(v,k)); fromdigits(Vec(v)) \\ Charles R Greathouse IV, Aug 02 2016

function dec2delta($k) {
  $p = -1;
  while ($k > 0) {
    $k -= $c = pow(2, floor(log($k, 2)));
    if ($p > -1) $d[] = $p - floor(log($c, 2));
    $p = floor(log($c, 2));
  }
  $d[] = $p + 1;
  return array_reverse($d);
}
function delta2dec($d) {
  $k = 0;
  $e = -1;
  foreach ($d AS $v) {
    if ($v > 0) {
      $e += $v;
      $k += pow(2, $e);
    }
  }
  return $k;
}

For n=1..511, a(n) = A004086(A004719(A071160(n))) [In other words, terms of A071160 with 0-digits deleted and the remaining digits reversed.] - Antti Karttunen, Sep 03 2016

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A214949 Numerator of sum of reciprocals of all nonzero digits of n in decimal representation.

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A214950 Denominator of sum of reciprocals of all nonzero digits of n in decimal representation.

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A243846 Numbers for which the nozero power-sequence of n falls into a loop.

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A125290 Numbers with at least two distinct digits in decimal representation, none of which is 0.

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A004177 Omit 2's from n.

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A243845 Numbers generated by recursive procedure a(n) = nozero(a(n-1) * 3), in which the function nozero(x) removes all zeros from x, starting with a(1) = 1.

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A247796 From right to left in decimal representation of n: replace each maximal set of adjacent digits with their sum, if this sum is less than 10.

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