A007954 Product of decimal digits of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,3

Comments

Moebius transform of A093811(n). a(n) = A093811(n) * A008683(n), where operation * denotes Dirichlet convolution, namely b(n) * c(n) = Sum_{d|n} b(d) * c(n/d). Simultaneously holds Dirichlet multiplication: a(n) * A000012(n) = A093811(n). - Jaroslav Krizek, Mar 22 2009

Apart from the 0's, all terms are in A002473. Further, for all m in A002473 there is some n such that a(n) = m, see A096867. - Charles R Greathouse IV, Sep 29 2013

a(n) = 0 asymptotically almost surely, namely for all n except for the set of numbers without digit '0'; this set is of density zero, since it is less and less probable to have no '0' as the number of digits of n grows. (See also A054054.) - M. F. Hasler, Oct 11 2015

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Rigoberto Flórez, Robinson A. Higuita and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

Florentin Smarandache, Only Problems, Not Solutions!.

Index entries for Colombian or self numbers and related sequences

Formula

A000035(a(A014261(n))) = 1. - Reinhard Zumkeller, Nov 30 2007

a(n) = abs(A187844(n)). - Reinhard Zumkeller, Mar 14 2011

a(n) > 0 if and only if A054054(n) > 0. a(n) = d in {1, ..., 9} if n = (10^k - 1)/9 + (d - 1)*10^m = A002275(k) + (d - 1)*A011557(m) for some k > m >= 0. The statement holds with "if and only if" for d in {1, 2, 3, 5, 7}. For d = 4, 6, 8 or 9, one has a(n) = d if n = (10^k - 1)/9 + (a - 1)*10^m + (b - 1)*10^p with integers k > m > p >= 0 and a, b > 0 such that d = a*b. - M. F. Hasler, Oct 11 2015

From Robert Israel, May 17 2016: (Start)

G.f.: Sum_{n >= 0} Product_{j = 0..n} Sum_{k = 1..9} k*x^(k*10^j).

G.f. satisfies A(x) = (x + 2*x^2 + ... + 9*x^9)*(1 + A(x^10)). (End)

a(n) <= 9^(1 + log_10(n/9)). - Lucas A. Brown, Jun 22 2023

Maple

A007954 := proc(n::integer)
    if n = 0 then
        0;
    else
        mul( d, d=convert(n, base, 10)) ;
    end if;
end proc: # R. J. Mathar, Oct 02 2019

Mathematica

Array[Times @@ IntegerDigits@ # &, 108, 0] (* Robert G. Wilson v, Mar 15 2011 *)

Prog

(PARI) A007954(n)= { local(resul = n % 10); n \= 10; while( n > 0, resul *= n %10; n \= 10; ); return(resul); } \\ R. J. Mathar, May 23 2006, edited by M. F. Hasler, Apr 23 2015
(PARI) A007954(n)=prod(i=1, #n=Vecsmall(Str(n)), n[i]-48) \\ (...eval(Vec(...)), n[i]) is about 50% slower; (...digits(n)...) about 6% slower. \\ M. F. Hasler, Dec 06 2009
(PARI) a(n)=if(n, factorback(digits(n)), 0) \\ Charles R Greathouse IV, Apr 14 2020
(Haskell)
a007954 n | n < 10 = n
          | otherwise = m * a007954 n' where (n', m) = divMod n 10
-- Reinhard Zumkeller, Oct 26 2012, Mar 14 2011
(Magma) [0] cat [&*Intseq(n): n in [1..110]]; // Vincenzo Librandi, Jan 03 2020
(Scala) (0 to 99).map(_.toString.toCharArray.map(_ - 48).scanRight(1)(_ * _).head) // Alonso del Arte, Apr 14 2020
(Python)
from math import prod
def a(n): return prod(map(int, str(n)))
print([a(n) for n in range(108)]) # Michael S. Branicky, Jan 16 2022

Crossrefs

Cf. A031347 (different from A035930), A007953, A007602, A010888, A093811, A008683, A000012, A061076 (partial sums), A230099.

Cf. A051802 (ignoring zeros).

Sequence in context: A087471 A128212 A187844 * A079475 A081286 A080867

Adjacent sequences: A007951 A007952 A007953 * A007955 A007956 A007957

Keyword

nonn,base,easy,nice,hear

Author

R. Muller

Extensions

Error in term 25 corrected, Nov 15 1995

Status

approved