A051801 -id:A051801 - cp's OEIS frontend

A127482 Product of the nonzero digital products of all the prime numbers prime(1) to prime(n).

2, 6, 30, 210, 210, 630, 4410, 39690, 238140, 4286520, 12859560, 270050760, 1080203040, 12962436480, 362948221440, 5444223321600, 244990049472000, 1469940296832000, 61737492466944000, 432162447268608000, 9075411392640768000, 571750917736368384000

Offset: 1

Alain Van Kerckhoven (alain(AT)avk.org), Sep 12 2007

			a(7) = dp_10(2)*dp_10(3)*dp_10(5)*dp_10(7)*dp_10(11)*dp_10(13)*dp_10(17) = 2*3*5*7*(1*1)*(1*3)*(1*7) = 4410.

Harvey P. Dale, Table of n, a(n) for n = 1..547

Cf. A000040, A007953, A007954, A051801, A101987, A131385, A131387, A131451.

a:= proc(n) option remember; `if`(n<1, 1, a(n-1)*mul(
     `if`(i=0, 1, i), i=convert(ithprime(n), base, 10)))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 11 2022

Rest[FoldList[Times,1,Times@@Cases[IntegerDigits[#],Except[0]]&/@ Prime[ Range[ 20]]]] (* Harvey P. Dale, Mar 19 2013 *)

f(n) = vecprod(select(x->(x>1), digits(prime(n)))); \\ A101987
a(n) = prod(k=1, n, f(k)); \\ Michel Marcus, Mar 11 2022

from math import prod
from sympy import sieve
def pod(s): return prod(int(d) for d in s if d != '0')
def a(n): return pod("".join(str(sieve[i+1]) for i in range(n)))
print([a(n) for n in range(1, 23)]) # Michael S. Branicky, Mar 11 2022

a(n) = Product_{k=1..n} dp_p(prime(k)) where prime(k)=A000040(k) and dp_p(m)=product of the nonzero digits of m in base p (p=10 for this sequence). - Hieronymus Fischer, Sep 29 2007
From Michel Marcus, Mar 11 2022: (Start)
a(n) = Product_{k=1..n} A051801(prime(k)).
a(n) = Product_{k=1..n} A101987(k). (End)

Corrected and extended by Hieronymus Fischer, Sep 29 2007

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.

Original entry on oeis.org

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

Derek Orr, Nov 25 2013

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.

			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.

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

Cf. A006753, A051801.

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 *)

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

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

Extended by T. D. Noe, Nov 28 2013

A335808 Nonzero multiplicative persistence in base 10: number of iterations of "multiply nonzero digits in base 10" needed to reach a number < 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 1, 2, 2, 3, 3, 2, 4, 3, 3, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 2, 3, 3, 3, 3, 3, 3, 2

Offset: 0

Lucas Colucci, Jun 24 2020

Coincides with A031346 up to n=204.

Differs from A087472 first at n=110. - R. J. Mathar, Aug 10 2020

R. K. Guy, Unsolved Problems in Number Theory, E16, pages 262-263.

Lucas Colucci, Table of n, a(n) for n = 0..9999
Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.

Cf. A031346, A051801.

A335808 := proc(n)
    option remember;
    if n < 10 then
        0 ;
    elif n < 20 then
        1 ;
    else
        A051801(n) ;
        1+procname(%) ;
    end if;
end proc: # R. J. Mathar, Aug 10 2020

Array[-1 + Length[NestWhileList[Times @@ DeleteCases[IntegerDigits[#], ?(# == 0 &)] &, #, # >= 10 &]] &, 105, 0] (* _Michael De Vlieger, Jun 24 2020 *)

a(n) = for (k=0, oo, if (n<10, return (k), n=vecprod(select(sign, digits(n))))) \\ Rémy Sigrist, Jul 18 2020

A338803 Product of the nonzero digits of (n written in base 5).

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 2, 2, 4, 6, 8, 3, 3, 6, 9, 12, 4, 4, 8, 12, 16, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 2, 2, 4, 6, 8, 3, 3, 6, 9, 12, 4, 4, 8, 12, 16, 2, 2, 4, 6, 8, 2, 2, 4, 6, 8, 4, 4, 8, 12, 16, 6, 6, 12, 18, 24, 8, 8, 16, 24, 32, 3, 3, 6, 9, 12, 3

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007091, A051801, A117592, A309956.

Table[Times @@ DeleteCases[IntegerDigits[n, 5], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4) A[x^5] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 5))); \\ Michel Marcus, Nov 12 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4) * A(x^5).

A338854 Product of the nonzero digits of (n written in base 4).

Original entry on oeis.org

1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 2, 2, 4, 6, 2, 2, 4, 6, 4, 4, 8, 12, 6, 6, 12, 18, 3, 3, 6, 9, 3, 3, 6, 9, 6, 6, 12, 18, 9, 9, 18, 27, 1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 1

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007090, A051801, A117592, A160382, A160383, A309954.

Table[Times @@ DeleteCases[IntegerDigits[n, 4], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3) A[x^4] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 4))); \\ Michel Marcus, Nov 12 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3) * A(x^4).

a(n) = 2^A160382(n) * 3^A160383(n).

A338863 Product of the nonzero digits of (n written in base 6).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 2, 4, 6, 8, 10, 3, 3, 6, 9, 12, 15, 4, 4, 8, 12, 16, 20, 5, 5, 10, 15, 20, 25, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 2, 4, 6, 8, 10, 3, 3, 6, 9, 12, 15, 4, 4, 8, 12, 16, 20, 5, 5, 10, 15, 20, 25, 2, 2, 4, 6, 8, 10, 2, 2, 4

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007092, A051801, A117592, A309957.

Table[Times @@ DeleteCases[IntegerDigits[n, 6], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5) A[x^6] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 6))); \\ Michel Marcus, Nov 13 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5) * A(x^6).

A338881 Product of the nonzero digits of (n written in base 8).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 2, 2, 4, 6, 8, 10, 12, 14, 3, 3, 6, 9, 12, 15, 18, 21, 4, 4, 8, 12, 16, 20, 24, 28, 5, 5, 10, 15, 20, 25, 30, 35, 6, 6, 12, 18, 24, 30, 36, 42, 7, 7, 14, 21, 28, 35, 42, 49, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 2

Offset: 0

Ilya Gutkovskiy, Nov 13 2020

Cf. A007094, A051801, A117592, A309959.

Table[Times @@ DeleteCases[IntegerDigits[n, 8], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5 + 6 x^6 + 7 x^7) A[x^8] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 8))); \\ Michel Marcus, Nov 14 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7) * A(x^8).

A342048 Numbers for which the sum of digits equals the product of nonzero digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 22, 30, 40, 50, 60, 70, 80, 90, 100, 123, 132, 200, 202, 213, 220, 231, 300, 312, 321, 400, 500, 600, 700, 800, 900, 1000, 1023, 1032, 1124, 1142, 1203, 1214, 1230, 1241, 1302, 1320, 1412, 1421, 2000, 2002, 2013, 2020, 2031, 2103, 2114, 2130, 2141, 2200

Offset: 1

Ilya Gutkovskiy, Feb 26 2021

			2103 is in the sequence because 2 + 1 + 0 + 3 = 2 * 1 * 3 = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007953, A034710, A051801.

q:= n-> (l-> is(add(i, i=l)=mul(i, i=l)))(
        subs(0=[][], convert(n, base, 10))):
select(q, [$0..3300])[];  # Alois P. Heinz, Feb 26 2021
# alternative
G:= proc(d,dmax,s,p) option remember; local i;
   if s + dmax*d < p or s > dmax^d*p then return [] fi;
   if d = 0 then return [[]] fi;
   [seq(op(map(t -> [i,op(t)], procname(d-1,i,s+i,p*i))),i=1..dmax)]
end proc:
f:= proc(d) local R,k,i;
  R:= [seq(op(map(t -> [0$k,op(t)], G(d-k,9,0,1))),k=0..d-1)];
  R:= map(op@combinat:-permute,R);
  sort(map(t -> add(t[i]*10^(i-1),i=1..d),R))
end proc:
f(4); # Robert Israel, Feb 28 2021

Select[Range[2200], Plus @@ IntegerDigits[#] == Times @@ DeleteCases[IntegerDigits[#], 0] &]

isok(k) = my(d=select(x->(x>0), digits(k))); vecprod(d) == vecsum(d); \\ Michel Marcus, Feb 26 2021

from math import prod
def ok(n):
  digs = list(map(int, str(n)))
  return sum(digs) == prod([d for d in digs if d != 0])
def aupto(lim): return [m for m in range(1, lim+1) if ok(m)]
print(aupto(2200)) # Michael S. Branicky, Feb 26 2021

A357526 Number of nonnegative integers less than n with the same product of the nonzero decimal digits as n.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 0, 0, 0, 0, 2, 3, 3, 2, 1, 0, 1, 0, 0, 0, 3, 4, 3, 2, 1, 0, 1, 0, 0, 0, 2, 3, 1, 1, 1, 0, 0, 0, 0, 0, 4, 5, 3, 2, 2, 1, 1, 0, 0, 0, 2, 3, 1, 1, 1, 1, 1, 0, 0, 0, 4, 5, 2, 3, 1, 1, 1, 1, 0, 0, 3

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

			a(1) = 1 because A051801(1) = 1 and also A051801(0) = 1.
a(21) = 3 because A051801(21) = 2 and also A051801(2) = A051801(12) = A051801(20) = 2.

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A051801, A138471, A254524, A338505.

Table[Length[Select[Range[0, n - 1], Times @@ DeleteCases[IntegerDigits[#], 0] == Times @@ DeleteCases[IntegerDigits[n], 0] &]], {n, 0, 90}]

a(n) = |{j < n : A051801(j) = A051801(n)}|.

A360075 a(n) is the product of the digits of A007602(n), the n-th Zuckerman number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 8, 18, 1, 2, 5, 16, 6, 15, 16, 35, 4, 12, 16, 6, 15, 96, 24, 12, 48, 84, 105, 48, 1, 2, 3, 5, 6, 3, 42, 32, 63, 4, 108, 3, 18, 48, 24, 175, 35, 4, 32, 24, 108, 3, 18, 144, 21, 252, 18, 135, 8, 64, 96, 96, 288, 108, 14, 63

Offset: 1

Rémy Sigrist, Jan 24 2023

Zuckerman numbers (A007602) correspond to numbers divisible by the product of their digits.

			For n = 1515:
- A007602(1515) = 11834112,
- so a(1515) = 1*1*8*3*4*1*1*2 = 192.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A007602, A007954, A051801, A288069, A325454.

{ for (n=1, 7119, p=vecprod(digits(n)); if (p && n%p==0, print1 (p", "))) }

a(n) = A007954(A007602(n)).
a(n) = A051801(A007602(n)).
a(n) * A288069(n) = A007602(n).

cp's OEIS Frontend

A127482 Product of the nonzero digital products of all the prime numbers prime(1) to prime(n).

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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.

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A335808 Nonzero multiplicative persistence in base 10: number of iterations of "multiply nonzero digits in base 10" needed to reach a number < 10.

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A338803 Product of the nonzero digits of (n written in base 5).

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A338854 Product of the nonzero digits of (n written in base 4).

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A338863 Product of the nonzero digits of (n written in base 6).

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A338881 Product of the nonzero digits of (n written in base 8).

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A342048 Numbers for which the sum of digits equals the product of nonzero digits.

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