A007954 -id:A007954 - cp's OEIS frontend

A054054 Smallest digit of n.

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

Offset: 0

Henry Bottomley, Apr 29 2000

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, i.e., except for a set of density 0: As the number of digits of n grows, the probability of having no zero digit goes to zero as 0.9^(length of n), although there are infinitely many counterexamples. - M. F. Hasler, Oct 11 2015

			a(12) = 1 because 1 < 2.

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

Cf. A054055.

Cf. A061601, A011540, A052382, A262188.

a054054 = f 9 where
   f m x | x <= 9 = min m x
         | otherwise = f (min m d) x' where (x',d) = divMod x 10
-- Reinhard Zumkeller, Jun 20 2012, Apr 25 2012

seq(min(convert(n,base,10)),n=0..100); # Robert Israel, Jul 07 2016

Mathematica
```
A054054[n_]:=Min[IntegerDigits[n]]
```

A054054(n)=if(n,vecmin(digits(n)))  \\ or: Set(digits(n))[1]. - M. F. Hasler, Jan 23 2013

a(A011540(n)) = 0; a(A052382(n)) > 0. - Reinhard Zumkeller, Apr 25 2012
a(n) = A262188(n,0). - Reinhard Zumkeller, Sep 14 2015
a(n) = 0 iff A007954(n) = 0. - M. F. Hasler, Oct 11 2015
a(n) = 9 - A054055(A061601(n)). - Robert Israel, Jul 07 2016

Edited by M. F. Hasler, Oct 11 2015

A048896 a(n) = 2^(A000120(n+1) - 1), n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4

Offset: 0

Wolfdieter Lang

a(n) = 2^A048881 = 2^{maximal power of 2 dividing the n-th Catalan number (A000108)}. [Comment corrected by N. J. A. Sloane, Apr 30 2018]
Row sums of triangle A128937. - Philippe Deléham, May 02 2007
a(n) = sum of (n+1)-th row terms of triangle A167364. - Gary W. Adamson, Nov 01 2009
a(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2, A117972(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - Daniel Forgues, Oct 16 2011
For n > 0: a(n) = A007954(A007931(n)). - Reinhard Zumkeller, Oct 26 2012
a(n) = A261363(2*(n+1), n+1). - Reinhard Zumkeller, Aug 16 2015
From Gus Wiseman, Oct 30 2022: (Start)
Also the number of coarsenings of the (n+1)-th composition in standard order. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See link for sequences related to standard compositions. For example, the a(10) = 4 coarsenings of (2,1,1) are: (2,1,1), (2,2), (3,1), (4).
Also the number of times n+1 appears in A357134. For example, 11 appears at positions 11, 20, 33, and 1024, so a(10) = 4.
(End)

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,2;
  1,2,2,4;
  1,2,2,4,2,4,4,8;
  1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16;
  1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32;
  ...,
the first half-rows converge to Gould's sequence A001316.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Transactions (2021) Vol. 1, No. 1 Article 1.
Neil J. Calkin, Eunice Y. S. Chan, Robert M. Corless, David J. Jeffrey, and Piers W. Lawrence, A Fractal Eigenvector, arXiv:2104.01116 [math.DS], 2021.
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
OEIS Wiki, Montgomery's pair correlation conjecture
Gus Wiseman, Statistics, classes, and transformations of standard compositions

This is Guy Steele's sequence GS(3, 5) (see A135416).
Equals first right hand column of triangle A160468.
Equals A160469(n+1)/A002425(n+1).
Cf. A000079, A001316, A117972, A160476, A167364, A220466, A261363.
Standard compositions are listed by A066099.
The opposite version (counting refinements) is A080100.
The version for Heinz numbers of partitions is A317141.
Cf. A000120, A001511, A029931, A048793, A058891, A070939, A272919, A357134.

a048896 n = a048896_list !! n
a048896_list = f [1] where f (x:xs) = x : f (xs ++ [x,2*x])
-- Reinhard Zumkeller, Mar 07 2011

import Data.List (transpose)
a048896 = a000079 . a000120
a048896_list = 1 : concat (transpose
   [zipWith (-) (map (* 2) a048896_list) a048896_list,
    map (* 2) a048896_list])
-- Reinhard Zumkeller, Jun 16 2013

[Numerator(2^n / Factorial(n+1)): n in [0..100]]; // Vincenzo Librandi, Apr 12 2014

a := n -> 2^(add(i,i=convert(n+1,base,2))-1): seq(a(n), n=0..97); # Peter Luschny, May 01 2009

NestList[Flatten[#1 /. a_Integer -> {a, 2 a}] &, {1}, 4] // Flatten (* Robert G. Wilson v, Aug 01 2012 *)
Table[Numerator[2^n / (n + 1)!], {n, 0, 200}] (* Vincenzo Librandi, Apr 12 2014 *)
Denominator[Table[BernoulliB[2*n] / (Zeta[2*n]/Pi^(2*n)), {n, 1, 100}]] (* Terry D. Grant, May 29 2017 *)
Table[Denominator[((2 n)!/2^(2 n + 1)) (-1)^n], {n, 1, 100}]/4 (* Terry D. Grant, May 29 2017 *)
2^IntegerExponent[CatalanNumber[Range[0,100]],2] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=if(n<1,1,if(n%2,a(n/2-1/2),2*a(n-1)))

a(n) = 1 << (hammingweight(n+1)-1); \\ Kevin Ryde, Feb 19 2022

a(n) = 2^A048881(n).
a(n) = 2^k if 2^k divides A000108(n) but 2^(k+1) does not divide A000108(n).
It appears that a(n) = Sum_{k=0..n} binomial(2*(n+1), k) mod 2. - Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 20 2001
a(0) = 1; a(2*n) = 2*a(2*n-1); a(2*n+1) = a(n).
a(n) = (1/2) * A001316(n+1). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
It appears that a(n) = Sum_{k=0..2n} floor(binomial(2n+2, k+1)/2)(-1)^k = 2^n - Sum_{k=0..n+1} floor(binomial(n+1, k)/2). - Paul Barry, Dec 24 2004
a(n) = Sum_{k=0..n} (T(n,k) mod 2) where T = A039598, A053121, A052179, A124575, A126075, A126093. - Philippe Deléham, May 02 2007
a(n) = numerator(b(n)), where sin(x)^2/x = Sum_{n>0} b(n)*(-1)^n x^(2*n-1). - Vladimir Kruchinin, Feb 06 2013
a((2*n+1)*2^p-1) = A001316(n), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 12 2013
a(n) = numerator(2^n / (n+1)!). - Vincenzo Librandi, Apr 12 2014
a(2n) = (2n+1)!/(n!n!)/A001803(n). - Richard Turk, Aug 23 2017
a(2n-1) = (2n-1)!/(n!(n-1)!)/A001790(n). - Richard Turk, Aug 23 2017

New definition from N. J. A. Sloane, Mar 01 2008

A051801 Product of the nonzero digits of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 4, 6, 8, 10, 12, 14, 16, 18, 3, 3, 6, 9, 12, 15, 18, 21, 24, 27, 4, 4, 8, 12, 16, 20, 24, 28, 32, 36, 5, 5, 10, 15, 20, 25, 30, 35, 40, 45, 6, 6, 12, 18, 24, 30, 36, 42, 48, 54, 7, 7, 14, 21

Offset: 0

Dan Hoey, Dec 09 1999

			a(0) = 1 since an empty product is 1 by convention. a(120) = 1*2 = 2.

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (First 1000 terms from Zak Seidov)
Index entries for Colombian or self numbers and related sequences

Basis for A051802.
See A338882 for similar sequences.
See also A007953 (digital sum).
Cf. A004719, A007954.

a051801 0 = 1
a051801 n = (a051801 n') * (m + 0 ^ m) where (n',m) = divMod n 10
-- Reinhard Zumkeller, Nov 23 2011

A051801 := proc(n) local d,j: d:=convert(n,base,10): return mul(`if`(d[j]=0,1,d[j]), j=1..nops(d)): end: seq(A051801(n),n=0..100); # Nathaniel Johnston, May 04 2011

(Times@@Cases[IntegerDigits[#],Except[0]])&/@Range[0,80] (* Harvey P. Dale, Jun 20 2011 *)
Table[Times@@(IntegerDigits[n]/.(0->1)),{n,0,80}] (* Harvey P. Dale, Apr 16 2023 *)

a(n)=my(v=select(k->k>1,digits(n)));prod(i=1,#v,v[i]) \\ Charles R Greathouse IV, Nov 20 2012

from operator import mul
from functools import reduce
def A051801(n):
    return reduce(mul, (int(d) for d in str(n) if d != '0')) if n > 0 else 1 # Chai Wah Wu, Aug 23 2014

from math import prod
def a(n): return prod(int(d) for d in str(n) if d != '0')
print([a(n) for n in range(74)]) # Michael S. Branicky, Jul 18 2021

// Swift 5
A051801(n): String(n).compactMap{$0.wholeNumberValue == 0 ? 1 : $0.wholeNumberValue}.reduce(1, *) // Egor Khmara, Jan 15 2021

a(n) = 1 if n=0, otherwise a(floor(n/10)) * (n mod 10 + 0^(n mod 10)). - Reinhard Zumkeller, Oct 13 2009
G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9) * A(x^10). - Ilya Gutkovskiy, Nov 14 2020
a(n) = A007954(A004719(n)). - Michel Marcus, Mar 07 2022

A034710 Positive numbers for which the sum of digits equals the product of digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 132, 213, 231, 312, 321, 1124, 1142, 1214, 1241, 1412, 1421, 2114, 2141, 2411, 4112, 4121, 4211, 11125, 11133, 11152, 11215, 11222, 11251, 11313, 11331, 11512, 11521, 12115, 12122, 12151, 12212, 12221, 12511

Offset: 1

Erich Friedman

Positive numbers k such that A007953(k) = A007954(k).
If k is a term, the digits of k are solutions of the equation x1*x2*...*xr = x1 + x2 + ... + xr; xi are from [1..9]. Permutations of digits (x1,...,xr) are different numbers k with the same property A007953(k) = A007954(k). For example: x1*x2 = x1 + x2; this equation has only 1 solution, (2,2), which gives the number 22. x1*x2*x3 = x1 + x2 + x3 has a solution (1,2,3), so the numbers 123, 132, 213, 231, 312, 321 have the property. - Ctibor O. Zizka, Mar 04 2008
Subsequence of A249334 (numbers for which the digital sum contains the same distinct digits as the digital product). With {0}, complement of A249335 with respect to A249334. Sequence of corresponding values of A007953(a(n)) = A007954(a(n)): 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ... contains only numbers from A002473. See A248794. - Jaroslav Krizek, Oct 25 2014
There are terms of the sequence ending in any term of A052382. - Robert Israel, Nov 02 2014
The number of digits which are not 1 in a(n) is O(log log a(n)) and tends to infinity as a(n) does. - Robert Dougherty-Bliss, Jun 23 2020

			1124 is a term since 1 + 1 + 2 + 4 = 1*1*2*4 = 8.

Alois P. Heinz, Table of n, a(n) for n = 1..27597 (first 1200 terms from T. D. Noe)

Cf. A002473, A007953, A007954, A052382, A061672, A248794, A249334, A249335.

Cf. A066306 (prime terms), A066307 (nonprimes).

import Data.List (elemIndices)
a034710 n = a034710_list !! (n-1)
a034710_list = elemIndices 0 $ map (\x -> a007953 x - a007954 x) [1..]
-- Reinhard Zumkeller, Mar 19 2011

[n: n in [1..10^6] | &*Intseq(n) eq &+Intseq(n)] // Jaroslav Krizek, Oct 25 2014

Select[Range[12512], (Plus @@ IntegerDigits[ # ]) == (Times @@ IntegerDigits[ # ]) &] (* Alonso del Arte, May 16 2005 *)

is(n)=my(d=digits(n)); vecsum(d)==factorback(d) \\ Charles R Greathouse IV, Feb 06 2017

Corrected by Larry Reeves (larryr(AT)acm.org), Jun 27 2001

Definition changed by N. J. A. Sloane to specifically exclude 0, Sep 22 2007

A051802 Nonzero multiplicative digital root of n.

Original entry on oeis.org

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

Offset: 0

Dan Hoey, Dec 09 1999

Occasionally defined with a(0) = 0.

Discussed Jun 15 1991 on sci.math by Mayne, Rusin, Landrum et al.

Uses A051801.

Cf. A007954.

a051802 = until (< 10) a051801  -- Reinhard Zumkeller, Nov 23 2011

A051801 := proc(n) local d,j: d:=convert(n,base,10): return mul(`if`(d[j]=0,1,d[j]), j=1..nops(d)): end: A051802 := proc(n) local m: if(n=0)then return 1:fi: m:=n: while(length(m)>1)do m:=A051801(m): od: return m: end: seq(A051802(n),n=0..100); # Nathaniel Johnston, May 04 2011

mdr0[n_] := NestWhile[Times @@ (IntegerDigits@# /. 0 -> 1) &, n, UnsameQ, All]; Table[ mdr0@n, {n, 0, 104}] (* Robert G. Wilson v, Aug 04 2006 *)

A051801(n)=my(v=select(k->k>1,digits(n)));prod(i=1,#v,v[i])
a(n)=while(n>9,n=A051801(n)); n \\ Charles R Greathouse IV, Nov 20 2012

from operator import mul
from functools import reduce
def A051802(n):
    if n == 0:
        return 1
    while n > 9:
        n = reduce(mul, (int(d) for d in str(n) if d != '0'))
    return n
# Chai Wah Wu, Aug 23 2014

def zeroLessIterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => zeroLessIterDigitProd(n.toString.replace("0", "").toCharArray.map( - 48).scanRight(1)( * ).head)
} // Note that zeroLessIterDigitProd(0) gives 0, not 1
List(1) ++: (1 to 99).map(zeroLessIterDigitProd) // Alonso del Arte, Apr 19 2020

If n == A051801(n) then n else a(A051801(n)).

More terms from Robert G. Wilson v, Aug 04 2006

A053666 Product of digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 6, 18, 3, 21, 4, 12, 28, 15, 45, 6, 42, 7, 21, 63, 24, 72, 63, 0, 0, 0, 0, 3, 14, 3, 21, 27, 36, 5, 35, 18, 42, 21, 63, 8, 9, 27, 63, 81, 2, 12, 28, 36, 18, 54, 8, 10, 70, 36, 108, 14, 98, 16, 48, 54, 0, 3, 9, 21, 9, 63, 84, 108, 45, 135, 126, 63, 189, 72, 216, 189

Offset: 1

Enoch Haga, Feb 16 2000

			a(25) = 63 because the 25th prime is 97, and 9 * 7 = 63.
a(26) = 0 because the 26th prime is 101, and 1 * 0 * 1 = 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007954.

[&*Intseq(NthPrime(n)): n in [1..80]]; // Vincenzo Librandi, Sep 15 2014

a:= n-> mul(i, i=convert(ithprime(n), base, 10)):
seq(a(n), n=1..78);  # Alois P. Heinz, Mar 11 2022

Table[Times@@IntegerDigits[Prime[n]], {n, 80}] (* Alonso del Arte, Feb 28 2014 *)

a(n) = {d = digits(prime(n), 10); return (prod(i=1, #d, d[i]));} \\ Michel Marcus, Jun 12 2013

from math import prod
from sympy import sieve
def pod(n): return prod(map(int, str(n)))
def a(n): return pod(sieve[n])
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Mar 11 2022

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)

A099542 Rhonda numbers to base 10.

Original entry on oeis.org

1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284, 25662, 33475, 34935, 35581, 45951, 47265, 47594, 52374, 53176, 53742, 54479, 55272, 56356, 56718, 95232, 118465, 133857, 148653, 154462, 161785

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 21 2004

An integer m is a Rhonda number to base b if the product of its digits in base b equals b*(sum of prime factors of m (taken with multiplicity)).
Does every Rhonda number to base 10 contain at least one 5? - Howard Berman (howard_berman(AT)hotmail.com), Oct 22 2008
Yes, every Rhonda number m to base 10 contains at least one 5 and also one even digit, otherwise A007954(m) mod 10 > 0. - Reinhard Zumkeller, Dec 01 2012

			1568 has prime factorization 2^5 * 7^2. Sum of prime factors = 2*5 + 7*2 = 24. Product of digits of 1568 = 1*5*6*8 = 240 = 10*24, hence 1568 is a Rhonda number to base 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (first 180 terms from Harvey P. Dale)
Kevin Brown, Infinitely Many Rhondas.
Giovanni Resta, Rhonda numbers the 64507 base-10 Rhonda numbers up to 10^12.
Walter Schneider, Rhonda Numbers.
Eric Weisstein's World of Mathematics, Rhonda Number.

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), A100970 (base 8), A100973 (base 9), A100971 (base 12), A100972 (base 14), A100974 (base 15), A100975 (base 16), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60), see also A255880.
Cf. A001414, A027746, A007954.
Column k=5 of A291925.

import Data.List (unfoldr); import Data.Tuple (swap)
a099542 n = a099542_list !! (n-1)
a099542_list = filter (rhonda 10) [1..]
rhonda b x = a001414 x * b == product (unfoldr
       (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b) x)
-- Reinhard Zumkeller, Mar 05 2015, Dec 01 2012

Select[Range[200000],10Total[Times@@@FactorInteger[#]]==Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Oct 16 2011 *)

A007954(a(n)) = 10 * A001414(a(n)).

A131451 Product of the nonzero digital products of all the numbers 1 to n (a 'total digital-product factorial' in base 10).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 362880, 362880, 725760, 2177280, 8709120, 43545600, 261273600, 1828915200, 14631321600, 131681894400, 263363788800, 526727577600, 2106910310400, 12641461862400, 101131694899200

Offset: 1

Hieronymus Fischer, Jul 11 2007

			a(12)=dp_10(1)*dp_10(2)*dp_10(3)*...*dp_10(11)*dp_10(12)=1*2*3*4*5*6* 7*8*9*1*(1*1)*(1*2).
a(345)=3*4*5*3^45*4^5*(3-1)!^100*(4-1)!^10*(5-1)!^1*9!^64.
a(1000)=9!^300. a(1111)=9!^321.

Hieronymus Fischer, Table of n, a(n) for n = 1..102

Cf. A131385, A010786, A131387, A007953, A007954, A037131.

with transforms;
f:=proc(n) option remember; if n = 0 then 1 else f(n-1)*digprod0(n); fi; end;[seq(f(n),n=0..40)]; # N. J. A. Sloane, Oct 12 2013

The following formulas are given for general bases p>1:

a(n)=product{1<=k<=n, dp_p(k)} where dp_p(k) = product of the nonzero digits of k in base p.

a(n)=(n mod p)!*product{00}(floor(n/p^j)mod p)^(1+(n mod p^j))*((floor(n/p^j)mod p)-1)!^(p^j).

Recurrence: a(n+k*p^m)=a(n)*k^n*a(k*p^m) for 0<=k

a(n)=n!, for 0<=n

a(k*p^m)=k*(p-1)!^(k*m*p^(m-1))*(k-1)!^(p^m) for 0<=k

a(n)=(p-1)!^((m*p^(m+1)-(m+1)*p^m+1)/(p-1)^2)=(p-1)!^(1+2*p+3*p^2+...+m*p^(m-1)) for n=1+p+p^2+...+p^m.

a(n)=(p-1)!^(k*(m*p^(m+1)-(m+1)*p^m+1)/(p-1)^2)*(k-1)!^(p*(p^m-1)/(p-1))*k^(k*(p^(m+1)-(m+1)*p+m)/(p-1)^2)*k!*k^m, for n=k*(1+p+p^2+...+p^m).

For p=10: a(10^n)=9!^(n*10^(n-1)).

Asymptotic behavior: a(10^n)=10^(0.5559763...*n*10^n). Hence it grows slower than the factorial A000142(10^n) for which we have (10^n)!=10^((n-0.43429448...)*10^n+n/2+0.3990899...+o(1/n)). Example: a(1000) has 1668 digits, whereas 1000! has 2568 digits.

New b-file from Hieronymus Fischer, Sep 10 2007

2 typos in the formula section removed by Hieronymus Fischer, Dec 05 2011

A028846 Numbers whose product of digits is a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 18, 21, 22, 24, 28, 41, 42, 44, 48, 81, 82, 84, 88, 111, 112, 114, 118, 121, 122, 124, 128, 141, 142, 144, 148, 181, 182, 184, 188, 211, 212, 214, 218, 221, 222, 224, 228, 241, 242, 244, 248, 281, 282, 284, 288, 411, 412, 414, 418, 421, 422, 424, 428, 441, 442, 444, 448

Offset: 1

N. J. A. Sloane

Numbers using only digits 1, 2, 4, and 8. - Michel Lagneau, Dec 01 2010

			28 is in the sequence because 2*8 = 2^4. - _Michel Lagneau_, Dec 01 2010

Bo Gyu Jeong, Table of n, a(n) for n = 1..5000

Cf. A007954, A028889, A028838.

Cf. A174813, A061426.

a028846 n = a028846_list !! (n-1)
a028846_list = f [1] where
   f ds = foldr (\d v -> 10 * v + d) 0 ds : f (s ds)
   s [] = [1]; s (8:ds) = 1 : s ds; s (d:ds) = 2*d : ds
-- Reinhard Zumkeller, Jan 13 2014

Select[Range[1000], IntegerQ[Log[2, Times @@ (IntegerDigits[#])]] &] (* Michel Lagneau, Dec 01 2010 *)

is(n)=#setminus(Set(digits(n)), [1,2,4,8])==0 \\ Charles R Greathouse IV, Apr 24 2025

from itertools import count, islice, product
def agen(): yield from (int("".join(p)) for d in count(1) for p in product("1248", repeat=d))
print(list(islice(agen(), 64))) # Michael S. Branicky, Aug 21 2022

def A028846(n):
    m = (k:=3*n+1).bit_length()-1>>1
    return sum(10**j<<((k-(1<<(m<<1)))//(3<<(j<<1))&3) for j in range(m)) # Chai Wah Wu, Jun 28 2025

Given a(0) = 0 and n = 4k - r, where 0 <= r <= 3, a(n) = 10*a(k-1) + 2^(3-r). - Clinton H. Dan, Aug 21 2022

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

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