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

Author

R. Muller

Keywords

• nonn
• base
• easy
• nice
• hear

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

Crossrefs

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

Cf. A051802 (ignoring zeros).

Programs

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

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 *)

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

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

Scala

(0 to 99).map(.toString.toCharArray.map( - 48).scanRight(1)( * ).head) // Alonso del Arte, Apr 14 2020

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

Extensions

Error in term 25 corrected, Nov 15 1995