A051801 -id:A051801 - cp's OEIS frontend

A087304 Smallest nontrivial multiple of n whose nonzero digit product is the same as that of the nonzero digit product of n. By nontrivial one means a(n) is not equal to n or (10^k)*n. 0 if no such number exists.

11, 12, 1113, 104, 15, 132, 1071, 24, 11133, 110, 1001, 1020, 1131, 1022, 105, 32, 1071, 108, 133, 120, 100002, 1122, 161, 1008, 125, 1430, 702, 224, 3016, 11130, 10013, 160, 3003, 612, 315, 1332, 703, 342, 11193, 1040, 10004, 1008, 602, 1144, 225

Offset: 1

Amarnath Murthy, Sep 01 2003

Conjecture: no term is zero.

			a(19) = 133 = 19*7 and 1*3*3 = 1*9.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A051801.

prd(n) = {my(d = digits(n), p = 1); for (k=1, #d, if (d[k], p *= d[k]);); p;}
a(n) = {my(k = 2, prdn = prd(n)); while (prd(k*n) != prdn, k++; if (! (k % 10), k++)); k*n;} \\ Michel Marcus, Aug 12 2017

from functools import reduce
from operator import mul
def A087304(n):
    i, p = 2, reduce(mul,(int(d) for d in str(n) if d != '0'))
    while (max(str(i)) == '1' and str(i).count('1') == 1) or reduce(mul,(int(d) for d in str(i*n) if d != '0')) != p:
        i += 1
    return i*n # Chai Wah Wu, Aug 11 2017

a(10*n) = a(n)*10. - Chai Wah Wu, Aug 11 2017

More terms from David Wasserman, Apr 19 2005

A233692 The smallest prime that produces a set of n primes such that every prime after the first one is equal to the previous plus the product of its nonzero digits.

Original entry on oeis.org

2, 23, 23, 239, 239, 239, 3413, 14249, 524219, 4167379, 324550981, 2589767209, 346333812907

Offset: 1

Carlos Rivera, Dec 14 2013

This sequence was produced as solution to the problem 1270 of Claudio Meller's website (see link).

			For n=3, initial prime=23, set {23, 29, 47} because 23 -> 23+2*3=29 -> 29+2*9=47.
For n=2 to 9, the sets are
      23,     29;
      23,     29,     47;
     239,    293,    347,    431;
     239,    293,    347,    431,    443;
     239,    293,    347,    431,    443,    491;
    3413,   3449,   3881,   4073,   4157,   4297,   4801;
   14249,  14537,  14957,  16217,  16301,  16319,  16481,  16673;
  524219, 524939, 534659, 550859, 559859, 640859, 649499, 719483, 725531.

Claudio Meller, 1270 - Usando el producto digital II, Números y algos mas (in Spanish).

Cf. A051801, A063114, A233783.

checkp(p, n) = {ok = isprime(p); for (i=1, n, print1(p, ", "); digs = digits(p); np = p + prod(i=1, #digs, if (d=digs[i], d, 1)); p = np;if (i != n, ok = ok && isprime(p));); ok;} \\ Michel Marcus, Dec 15 2013

a(13) from Giovanni Resta, Dec 15 2013

A243140 Numbers n such that n appears in the sequence beginning with the digit-product of n and extended by adding successive digit-products.

Original entry on oeis.org

22, 26, 38, 55, 62, 88, 95, 102, 104, 108, 116, 122, 126, 138, 162, 174, 202, 206, 218, 234, 258, 410, 414, 430, 442, 474, 586, 826, 922, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 3174, 3258, 3498, 4362

Offset: 1

Anthony Sand, May 30 2014

Numbers n>9 with following property: form a sequence b(i) whose initial term is digit-product(n). Later terms are given by the rule that b(i) = b(i-1) + digit-product(b(i-1)) and n itself appears in the sequence.

The function digit-product(n) multiplies all nonzero digits of n (A051801). For example, digit-product(1230) = 1 * 2 * 3 = 6. The resultant sequence appears in A063114, n + product of nonzero digits of n.

			The digit-product sequence for 22 begins with digit-product(22)= 4, 4 + 4 = 8, 8 + 8 = 16, 16 + 6 = 22. Since this procedure returns to the initial number 22, it belongs here.
The digit-product sequence for 102 begins with 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16, 16 + 6 = 22, 22 + 4 = 26, 26 + 12 = 38, 38 + 24 = 62, 62 + 12 = 74, 74 + 28 = 102. Since this procedure returns to the initial number 102, it belongs here.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Cf. A007629, A243021, A051801, A063114.

dp(n)=my(v=select(k->k>1,digits(n))); prod(i=1,#v,v[i])
is(n)=my(t=dp(n)); until(t>=n, t+=dp(t)); t==n \\ Charles R Greathouse IV, Jun 05 2014

b(i) = b(i-1) + digit-product(b(i-1)).

A338880 Product of the nonzero digits of (n written in base 7).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 2, 2, 4, 6, 8, 10, 12, 3, 3, 6, 9, 12, 15, 18, 4, 4, 8, 12, 16, 20, 24, 5, 5, 10, 15, 20, 25, 30, 6, 6, 12, 18, 24, 30, 36, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 2, 2, 4, 6, 8, 10, 12, 3, 3, 6, 9, 12, 15, 18, 4, 4, 8, 12

Offset: 0

Ilya Gutkovskiy, Nov 13 2020

Alois P. Heinz, Table of n, a(n) for n = 0..9603

Cf. A007093, A051801, A053828, A117592, A309958.

Table[Times @@ DeleteCases[IntegerDigits[n, 7], 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) A[x^7] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 7))); \\ 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) * A(x^7).

A351327 Numbers whose trajectory under iteration of the product of squares of nonzero digits map includes 1.

Original entry on oeis.org

1, 5, 10, 11, 15, 25, 50, 51, 52, 100, 101, 105, 110, 111, 115, 125, 150, 151, 152, 205, 215, 250, 251, 255, 357, 375, 455, 500, 501, 502, 510, 511, 512, 520, 521, 525, 537, 545, 552, 554, 573, 735, 753, 1000, 1001, 1005, 1010, 1011, 1015, 1025, 1050, 1051

Offset: 1

Luca Onnis, Feb 07 2022

To determine whether a given number k is a term of this sequence, start with k, take the square of the product of its nonzero digits, apply the same process to the result, and continue until 1 is reached or a loop is entered. If 1 is reached, k is a term of this sequence.
Every power 10^k is a term of this sequence.
If k is a term, the numbers obtained by inserting zeros anywhere in k are terms.
If k is a term, the numbers obtained by inserting ones anywhere in k are terms.
If k is a term, each distinct permutation of the digits of k gives another term.
If k is a term, the number of iterations required to converge to 1 is less than or equal to 3 (conjectured).
From Michael S. Branicky, Feb 07 2022: (Start)
The product of squares of nonzero digits map, f, has fixed points given in A115385.
The map f has (at least) the following cycles:
  - 324, 576, 44100, 256, 3600;
  - 11664, 20736, 63504, 129600;
  - 15876, 2822400, 65536, 7290000;
  - 5308416, 8294400;
  - 49787136000000, 64524128256, 849346560000, 386983526400, 55725627801600.
(End)

			255 is a term of the sequence: the square of the product of its nonzero digits is (2*5*5)^2=2500, the square of the product of its nonzero digits is (2*5)^2=100, and the square of the product of its nonzero digits is 1^2=1.
2 is not a term of the sequence because its trajectory under the map is 2 -> 4 -> 16 -> 36 -> 324 -> 576 -> 44100 -> 256 -> 3600 -> 324 (reached earlier), so it enters a loop and never reaches 1.

Luca Onnis, On a variant of the happy numbers and their generalizations, arXiv:2203.03381 [math.GM], 2022.

Cf. A007770, A051801, A115385.

b:= proc() false end:
q:= proc(n) local m, s; m, s:= n, {};
      do if m=1 then return true
       elif m in s or b(m) then b(n):= true; return false
       else s, m:= {s[], m}, mul(max(1, i)^2, i=convert(m, base, 10))
         fi
      od
    end:
select(q, [$1..2000])[];  # Alois P. Heinz, Feb 11 2022

Select[Range[1000],
FixedPoint[
    Product[ReplaceAll[0 -> 1][IntegerDigits[#]][[i]]^2, {i, 1,
       Length[ReplaceAll[0 -> 1][IntegerDigits[#]]]}] &, #, 10] == 1 &]

f(n) = vecprod(apply(d -> if (d, d^2, 1), digits(n)))
is(n) = { my (m=f(n)); while (1, if (n==1, return (1), n==m, return (0), n=f(n); m=f(f(m)))) } \\ Rémy Sigrist, Feb 11 2022

from math import prod
def psd(n): return prod(int(d)**2 for d in str(n) if d != "0")
def ok(n):
    seen = set()
    while n not in seen: # iterate until fixed point or in cycle
        seen.add(n)
        n = psd(n)
    return n == 1
def aupto(n): return [k for k in range(1, n+1) if ok(k)]
print(aupto(1205)) # Michael S. Branicky, Feb 07 2022

A382895 Divide n successively by its nonzero digits from most to least significant, updating the result at each step and skipping any digit that doesn't divide the current value exactly.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 6, 13, 14, 3, 16, 17, 18, 19, 10, 21, 11, 23, 3, 5, 13, 27, 14, 29, 10, 31, 16, 11, 34, 7, 2, 37, 38, 13, 10, 41, 21, 43, 11, 9, 46, 47, 12, 49, 10, 51, 26, 53, 54, 11, 56, 57, 58, 59, 10, 61, 31, 21, 16, 13, 11, 67, 68, 69, 10, 71, 36, 73, 74, 15

Offset: 1

Seiichi Manyama, Apr 08 2025

			36 is divisible by 3, so divide by 3 to get 12.
12 is divisible by 6, so divide by 6 to get  2. So a(36) = 2.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A051801, A382897.

def A(n):
    m = n
    for i in map(int, str(n)):
        if i != 0 and m % i == 0:
            m //= i
    return m
def A382895(n):
    return [A(i) for i in range(1, n + 1)]
print(A382895(100))

def A(n)
  m = n
  n.to_s.split('').map(&:to_i).each{|i|
    m /= i if i != 0 && m % i == 0
  }
  m
end
def A382895(n)
  (1..n).map{|i| A(i)}
end
p A382895(100)

A382897 a(n) = n / A382895(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 2, 1, 2, 1, 8, 5, 2, 1, 2, 1, 3, 1, 2, 3, 1, 5, 18, 1, 1, 3, 4, 1, 2, 1, 4, 5, 1, 1, 4, 1, 5, 1, 2, 1, 1, 5, 1, 1, 1, 1, 6, 1, 2, 3, 4, 5, 6, 1, 1, 1, 7, 1, 2, 1, 1, 5, 1, 7, 1, 1, 8, 1, 2, 1, 4, 5, 1, 1, 8, 1, 9, 1, 2, 3, 1, 5, 6, 1, 1, 9

Offset: 1

Seiichi Manyama, Apr 08 2025

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A051801, A382895.

def A(n):
    m = n
    for i in map(int, str(n)):
        if i != 0 and m % i == 0:
            m //= i
    return n // m
def A382897(n):
    return [A(i) for i in range(1, n + 1)]
print(A382897(100))

def A(n)
  m = n
  n.to_s.split('').map(&:to_i).each{|i|
    m /= i if i != 0 && m % i == 0
  }
  n / m
end
def A382897(n)
  (1..n).map{|i| A(i)}
end
p A382897(100)

A086355 Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 6, 8, 3, 2, 4, 2, 6, 5, 2, 6, 8, 7, 2, 8, 8, 4, 8, 1, 3, 7, 9, 3, 4, 3, 2, 4, 8, 5, 5, 8, 8, 2, 8, 8, 9, 4, 8, 8, 2, 2, 6, 8, 8, 2, 8, 1, 7, 8, 8, 4, 4, 6, 6, 2, 2, 3, 9, 2, 9, 8, 6, 8, 2, 5, 2, 8, 4, 4, 2, 4, 4, 8, 8, 8, 2, 8, 8, 6, 6, 4, 8, 4, 6, 2, 6, 8, 8, 5, 2, 1, 3, 2, 4, 5, 9, 4, 5

Offset: 1

Labos Elemer, Jul 21 2003

			n=100, prime(100)=541, iteration list={541,20,2}, a(100)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[prd, Prime[w]], {w, 1, 128}]

a(n) = A051802(A000040(n)) = fixed-point of A051801(n-th prime).

A086992 Product of nonzero digits in n-th row of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 9, 96, 25, 1800, 44100, 103219200, 3869835264, 128000, 104976000000, 3071875232563200, 7050692013745766400, 626913312768, 332150625000000000000, 1292730125539029811200

Offset: 1

Jason Earls, Jul 29 2003

			a(6) = 25 because the digits in the 5th row of Pascal's triangle are 1,5,(1,0),(1,0),5,1, the product of the nonzero terms is 25. - _Richard M. Green_, Feb 12 2014

Cf. A007318, A066600, A086993, A051801.

A086992[n_]:= Times @@ DeleteCases[0]@Flatten@IntegerDigits@Table[Binomial[n, k], {k, 0, n}] (* JungHwan Min, Dec 07 2015 *)

A051801(n)=my(v=select(k->k>1, digits(n))); prod(i=1, #v, v[i])
a(n)=prod(k=1,(n-1)\2,A051801(binomial(n,k)))^2*if(n%2,1,A051801(binomial(n,n/2))) \\ Charles R Greathouse IV, Dec 08 2015

A096301 Product of nonzero digits(sum of digits(n^n)).

Original entry on oeis.org

1, 4, 9, 3, 1, 14, 10, 21, 20, 1, 4, 20, 40, 10, 81, 64, 72, 8, 14, 3, 7, 32, 24, 15, 5, 20, 72, 18, 42, 18, 32, 10, 14, 8, 12, 14, 14, 60, 18, 2, 24, 15, 9, 30, 18, 30, 84, 27, 75, 5, 16, 8, 15, 168, 32, 27, 20, 56, 28, 24, 128, 20, 180, 224, 70, 15, 60, 120, 196, 90, 144, 200, 24

Offset: 1

Jason Earls, Jun 25 2004

Conjecture: a(n) = a(n+1) for infinitely many positive integers. Largest found is n=4462, i.e. pnd(sd(4462^4462)) = pnd(sd(4463^4463)) = 126.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051801, A007953.

Array[Times@@Select[IntegerDigits[Total[IntegerDigits[#^#]]],#>0&]&,80] (* Harvey P. Dale, Sep 30 2012 *)

cp's OEIS Frontend

A087304 Smallest nontrivial multiple of n whose nonzero digit product is the same as that of the nonzero digit product of n. By nontrivial one means a(n) is not equal to n or (10^k)*n. 0 if no such number exists.

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A233692 The smallest prime that produces a set of n primes such that every prime after the first one is equal to the previous plus the product of its nonzero digits.

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A243140 Numbers n such that n appears in the sequence beginning with the digit-product of n and extended by adding successive digit-products.

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PARI

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A338880 Product of the nonzero digits of (n written in base 7).

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Mathematica

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A351327 Numbers whose trajectory under iteration of the product of squares of nonzero digits map includes 1.

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Maple

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A382895 Divide n successively by its nonzero digits from most to least significant, updating the result at each step and skipping any digit that doesn't divide the current value exactly.

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A382897 a(n) = n / A382895(n).

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A086355 Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.

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