A232541 Multiplicative Smith numbers: Composite numbers n such that the product of nonzero digits of n = product of nonzero digits of prime factors of n.

4, 6, 8, 9, 95, 159, 195, 249, 326, 762, 973, 995, 998, 1057, 1086, 1111, 1189, 1236, 1255, 1337, 1338, 1383, 1389, 1395, 1419, 1509, 2139, 2248, 2623, 2679, 2737, 2928, 2949, 3029, 3065, 3202, 3344, 3345, 3419, 3432, 3437, 3464, 3706, 3945, 4344, 4502

Offset: 1

Author

Derek Orr, Nov 25 2013

Keywords

• nonn
• base
• easy

Comments

They follow the same formula for Smith numbers, however, instead of addition, we have multiplication (only nonzero digits are included).

Trivially, prime numbers satisfy this property but are not included in the sequence.

Examples

			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).
998 is a member of this sequence because 998 = 2*499 and 9*9*8 = 2*4*9*9.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006753, A051801.

Programs

Mathematica

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 *)
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 *)

Python

import sympy
from sympy import isprime
from sympy import factorint
def DigitProd(x):
    prod = 1
    for i in str(x):
        if i != '0':
            prod *= int(i)
    return prod
def f(x):
    lst = []
    for n in range(len(list(factorint(x)))):
        lst.append(str(list(factorint(x))[n])*list(factorint(x).values())[n])
    string = ''
    for i in lst:
        string += i
    prod = 1
    for a in string:
        if a != '0':
            prod *= int(a)
    if prod == DigitProd(x):
        return True
x = 4
while x < 10**3:
    if not isprime(x):
        if f(x):
            print(x)
    x += 1

Sage

def prodPrimeDig(x):
    F=factor(x)
    T=[item for sublist in [[y[0]]*y[1] for y in F] for item in sublist]
    return prod([prod(filter(lambda a: a!=0,h.digits(base=10))) for h in T])
n=3345 #Change n for more digits
[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

Extensions

Extended by T. D. Noe, Nov 28 2013