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%I #40 Jan 15 2022 15:15:30
%S 4,6,8,9,95,159,195,249,326,762,973,995,998,1057,1086,1111,1189,1236,
%T 1255,1337,1338,1383,1389,1395,1419,1509,2139,2248,2623,2679,2737,
%U 2928,2949,3029,3065,3202,3344,3345,3419,3432,3437,3464,3706,3945,4344,4502
%N Multiplicative Smith numbers: Composite numbers n such that the product of nonzero digits of n = product of nonzero digits of prime factors of n.
%C They follow the same formula for Smith numbers, however, instead of addition, we have multiplication (only nonzero digits are included).
%C Trivially, prime numbers satisfy this property but are not included in the sequence.
%H T. D. Noe, <a href="/A232541/b232541.txt">Table of n, a(n) for n = 1..1000</a>
%e 1236 is a member of this sequence because 1236 = 2*2*3*103 and 1*2*3*6 = 2*2*3*1*3 (zeros are not included).
%e 998 is a member of this sequence because 998 = 2*499 and 9*9*8 = 2*4*9*9.
%t f[n_] := Times @@ DeleteCases[IntegerDigits[n], 0]; pFactors[n_] := Module[{f = FactorInteger[n]}, Flatten[ConstantArray @@@ f]]; Select[Range[2, 10000], ! PrimeQ[#] && f[#] == Times @@ f /@ pFactors[#] &] (* _T. D. Noe_, Nov 28 2013 *)
%t msnQ[n_]:=Times@@(Flatten[IntegerDigits/@Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]/.(0->1))==Times@@(IntegerDigits[n]/.(0->1)); Select[ Range[ 5000],CompositeQ[#]&&msnQ[#]&] (* _Harvey P. Dale_, Jan 15 2022 *)
%o (Python)
%o import sympy
%o from sympy import isprime
%o from sympy import factorint
%o def DigitProd(x):
%o prod = 1
%o for i in str(x):
%o if i != '0':
%o prod *= int(i)
%o return prod
%o def f(x):
%o lst = []
%o for n in range(len(list(factorint(x)))):
%o lst.append(str(list(factorint(x))[n])*list(factorint(x).values())[n])
%o string = ''
%o for i in lst:
%o string += i
%o prod = 1
%o for a in string:
%o if a != '0':
%o prod *= int(a)
%o if prod == DigitProd(x):
%o return True
%o x = 4
%o while x < 10**3:
%o if not isprime(x):
%o if f(x):
%o print(x)
%o x += 1
%o (Sage)
%o def prodPrimeDig(x):
%o F=factor(x)
%o T=[item for sublist in [[y[0]]*y[1] for y in F] for item in sublist]
%o return prod([prod(filter(lambda a: a!=0,h.digits(base=10))) for h in T])
%o n=3345 #Change n for more digits
%o [k for k in [1..n] if prod(filter(lambda a: a!=0,k.digits(base=10)))==prodPrimeDig(k) and not(is_prime(k))] # _Tom Edgar_, Nov 26 2013
%Y Cf. A006753, A051801.
%K nonn,base,easy
%O 1,1
%A _Derek Orr_, Nov 25 2013
%E Extended by _T. D. Noe_, Nov 28 2013
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