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A007954 Product of decimal digits of n. 293
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 (list; graph; refs; listen; history; text; internal format)
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
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!.
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
KEYWORD
nonn,base,easy,nice,hear
AUTHOR
R. Muller
EXTENSIONS
Error in term 25 corrected, Nov 15 1995
STATUS
approved

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Last modified February 27 21:03 EST 2024. Contains 370378 sequences. (Running on oeis4.)