A067734 -id:A067734 - cp's OEIS frontend

A068191 Numbers n such that A067734(n)=0; complement of A002473; at least one prime-factor of n is larger than 7, it has 2 decimal digits.

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102, 103, 104, 106, 107, 109, 110, 111, 113, 114

Offset: 1

Labos Elemer, Feb 19 2002

Also numbers n such that A198487(n) = 0 and A107698(n) = 0. - Jaroslav Krizek, Nov 04 2011
A086299(a(n)) = 0. - Reinhard Zumkeller, Apr 01 2012
A262401(a(n)) < a(n). - Reinhard Zumkeller, Sep 25 2015
Numbers not in A007954. - Mohammed Yaseen, Sep 13 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A002473, A067734.
Cf. A068183, A068184, A068185, A068186, A068187, A068189, A068190.
Cf. A262401.

import Data.List (elemIndices)
a068191 n = a068191_list !! (n-1)
a068191_list = map (+ 1) $ elemIndices 0 a086299_list
-- Reinhard Zumkeller, Apr 01 2012

Select[Range@120, Last@Map[First, FactorInteger@#] > 7 &] (* Vincenzo Librandi, Sep 19 2016 *)

from sympy import integer_log
def A068191(n):
    def f(x):
        c = n
        for i in range(integer_log(x,7)[0]+1):
            i7 = 7**i
            m = x//i7
            for j in range(integer_log(m,5)[0]+1):
                j5 = 5**j
                r = m//j5
                for k in range(integer_log(r,3)[0]+1):
                    c += (r//3**k).bit_length()
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 16 2024

A068189 Smallest positive number whose product of digits equals n, or a(n)=0 if no such number exists, i.e. when n has a prime divisor greater than 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 0, 26, 0, 27, 35, 28, 0, 29, 0, 45, 37, 0, 0, 38, 55, 0, 39, 47, 0, 56, 0, 48, 0, 0, 57, 49, 0, 0, 0, 58, 0, 67, 0, 0, 59, 0, 0, 68, 77, 255, 0, 0, 0, 69, 0, 78, 0, 0, 0, 256, 0, 0, 79, 88, 0, 0, 0, 0, 0, 257, 0, 89, 0, 0, 355, 0, 0, 0, 0, 258, 99, 0, 0, 267, 0

Offset: 1

Labos Elemer, Feb 19 2002

a(n) > 0 if and only if n is in A002473.

			n=2,10,50,250 gives a(n)=2,25,255,2555; n=11,39,78, etc..a(n)=0.
10000 = 2 * 5 * 5 * 5 * 5 * 8. No product of two of these factors is less than 10 so a(10000) = 255558 (the concatenation of these factors in nondecreasing order). - _David A. Corneth_, Jul 31 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001222, A002473, A007954, A067734, A068183-A068187, A068189-A068191.

Cf. A085123, A280249.

f[x_] := Apply[Times, IntegerDigits[x]] a = Table[0, {256} ]; Do[ b = f[n]; If[b < 257 && a[[b]] == 0, a[[b]] =n], {n, 1, 10000} ]; a

a(n) = {if(n==1, return(1)); my(res = []); forstep(i=9,2,-1, v = valuation(n, i); if(v > 0, res = concat(vector(v, j, i), res); n/=i^v)); if(n==1,fromdigits(res), 0)} \\ David A. Corneth, Jul 31 2017

def convert(n):
    if n == 1:
        return 1
    result = 0
    cur = 1
    while n > 1:
        found = False
        for i in range(9, 1, -1):
            if n % i == 0:
                result += cur * i
                cur *= 10
                n //= i
                found = True
                break
        if not found:
            return 0
    return result
N = 256
for n in range(1, N):
    print(n, convert(n))
# Dmitry Kamenetsky, Oct 20 2008

A068183 Largest number without decimal digits equal to 1 whose product of digits gives n!.

Original entry on oeis.org

2, 32, 3222, 53222, 5332222, 75332222, 75332222222, 7533332222222, 755333322222222

Offset: 2

Labos Elemer, Feb 18 2002

Maximal solutions are obtained from the concatenation of distinct prime factors that have one decimal digit. The sequence is finite because, for n>10, n! has 2-digit prime factors.

Cf. A000142, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

a(n)=Max{x; f[x]=n!}, where x has no digit=1 and f[x_] := Apply[Times, IntegerDigits[x]].

A068187 a(n) is the smallest number such that the product of its decimal digits equals n^n, or 0 if no solutions exist.

Original entry on oeis.org

1, 4, 39, 488, 55555, 88999, 7777777, 88888888, 999999999, 25555555555888, 0, 88888888999999, 0, 4777777777777778888, 35555555555555559999999, 2888888888888888888888, 0, 888888999999999999999999, 0, 2555555555555555555558888888888888, 37777777777777777777779999999999

Offset: 1

Labos Elemer, Feb 18 2002

a(n) = 0 if and only if n has a prime factor > 7. If n > 1 has no prime factor > 7, let n^n = 2^a*3^b*5^c*7^d. Let m(x) denote the number of digit x in a(n). Then a(n) is a number whose digits are nondecreasing and defined as follows. m(2) = 1 if a mod 3 == 1 and 0 otherwise, m(3) = 1 if b mod 2 == 1 and 0 otherwise, m(4) = 1 if a mod 3 == 2 and 0 otherwise, m(5) = c, m(6) = 0, m(7) = d, m(8) = floor(a/3), m(9) = floor(b/2). - Chai Wah Wu, Aug 12 2017

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A000312, A067734, A068183, A068184, A068185, A068186, A068188, A068190.

from sympy import factorint
def A068187(n):
    if n == 1:
        return 1
    pf = factorint(n)
    return 0 if max(pf) > 7 else int(''.join(sorted(''.join(str(a)*(n*b) for a,b in pf.items()).replace('222','8').replace('22','4').replace('33','9')))) # Chai Wah Wu, Aug 13 2017

Edited by Dean Hickerson and Henry Bottomley, Mar 07 2002

A068184 Smallest number whose product of digits equals n!.

Original entry on oeis.org

1, 1, 2, 6, 38, 358, 2589, 25789, 257889, 2578879, 45578899

Offset: 0

Labos Elemer, Feb 18 2002

The sequence is finite because n! for n>10 has 2-digit prime factors.

			For n=4 the solutions having digit products equal 24 excluding those with digit 1 are: {38, 46, 64, 83, 226, 234, 243, 262, 324, 342, 423, 432, 622, 2223, 2232, 2322, 3222} of which the smallest is 38. For n>1, numbers with a digit 1 are too big.

Cf. A000142, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

Edited by Henry Bottomley, Feb 26 2002

A068185 Number of ways writing n^n as a product of decimal digits of some other number which has no digits equal to 1.

Original entry on oeis.org

0, 2, 3, 81, 1, 102136, 1, 1389537, 4181, 4972825, 0, 12718670252691776, 0, 4506838380, 11472991008, 53560898629395777, 0, 514875062240230100091396, 0, 164997736300578242823300, 241098942106440, 0, 0, 3203410440031870942324022423896806853153460, 1, 0, 61305790721611591

Offset: 1

Labos Elemer, Feb 19 2002

a(n)= 0 when n has prime-factor larger than 7 [so A067734(n)=0] or when n is in A068191, i.e. not in A002473.

			n=1 has no solution; a(2)=A000073(6)=2 with {4,22} solutions; a(3)=A067734(27)=3=Fibonacci[4]; n=5 and n=7, n^n has single prime factor of which any true multiple have 2 digits so 55555 and 7777777 are the only solutions, so a(5)=a(7)=1; a(4)=A067734(256)=81=A000073(10); a(8)=A067734(2^24)=A000073(26)=1389537; n=9 a(9)=A067734(3^27)=4181.

Cf. A000045, A000073, A000312, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

a(n) = A067734(n^n) = A067734(A000312(n))

Edited By Henry Bottomley, Feb 26 2002.
Edited and extended by Max Alekseyev, Sep 19 2009
a(9) corrected by Sean A. Irvine, Feb 01 2024

A068186 a(n) is the largest number whose product of decimal digits equals n^n.

Original entry on oeis.org

22, 333, 22222222, 55555, 333333222222, 7777777, 222222222222222222222222, 333333333333333333, 55555555552222222222, 0, 333333333333222222222222222222222222, 0, 7777777777777722222222222222

Offset: 2

Labos Elemer, Feb 19 2002

No digit=1 is permitted to avoid infinite number of solutions; a(n)=0 if A067734(n^n)=0.

			n=10, 10^10=10000000000, a(5)=55555555552222222222.

Chai Wah Wu, Table of n, a(n) for n = 2..100

Cf. A000312, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

from sympy import factorint
def A068186(n):
    if n == 1:
        return 1
    pf = factorint(n)
    ps = sorted(pf.keys(),reverse=True)
    if ps[0] > 7:
        return 0
    s = ''
    for p in ps:
        s += str(p)*(n*pf[p])
    return int(s) # Chai Wah Wu, Aug 12 2017

a(n) is obtained as prime factors of n^n concatenated in order of magnitude and with repetitions; a(n)=0 if n has p > 7 prime factors.

a(12) corrected by Chai Wah Wu, Aug 12 2017

A068190 Largest number whose digit product equals n; a(n)=0 if no such number exists, e.g., when n has a prime factor larger than 7; no digit=1 is permitted to avoid an infinite number of solutions.

Original entry on oeis.org

0, 2, 3, 22, 5, 32, 7, 222, 33, 52, 0, 322, 0, 72, 53, 2222, 0, 332, 0, 522, 73, 0, 0, 3222, 55, 0, 333, 722, 0, 532, 0, 22222, 0, 0, 75, 3322, 0, 0, 0, 5222, 0, 732, 0, 0, 533, 0, 0, 32222, 77, 552, 0, 0, 0, 3332, 0, 7222, 0, 0, 0, 5322, 0, 0, 733, 222222, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Feb 19 2002

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001222, A002473, A007954, A067734, A068183-A068187, A068189-A068191.

Array[If[#[[-1, 1]] > 7, 0, FromDigits@ Reverse@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, #]] &@ FactorInteger@ # &, 69] /. 1 -> 0 (* Michael De Vlieger, Dec 08 2018 *)

a(n) = {my(res = []); for(i=2, 9, v = valuation(n, i); if(v > 0, res = concat(vector(v, j, i), res); n/=i^v)); if(n==1,fromdigits(res), 0)} \\ David A. Corneth, Jul 31 2017

If a solution exists, a(n) is the concatenation of prime factors with repetitions and in order of magnitude, otherwise a(n)=0.

a(36) corrected by David A. Corneth, Jul 31 2017

A288738 Number of ways writing the n-th 7-smooth number as a product of decimal digits of some other number which has no digits equal to 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 1, 1, 4, 3, 2, 1, 5, 1, 2, 2, 2, 5, 1, 6, 3, 2, 2, 7, 1, 1, 4, 3, 3, 2, 7, 1, 9, 1, 4, 3, 3, 3, 9, 1, 2, 1, 7, 4, 5, 1, 3, 8, 2, 2, 13, 1, 2, 5, 5, 5, 1, 6, 2, 12, 2, 3, 2, 12, 5, 2, 7, 3, 1, 1, 6, 10, 4, 3, 17, 2, 3, 2, 7, 10, 7

Offset: 1

David A. Corneth, Jun 22 2017

This sequence is A067734 without the zeros.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A002473, A007602, A067734.

A069468 Number of ways writing n! as a product of some other numbers which has no digits equal to 1.

Original entry on oeis.org

0, 1, 3, 17, 68, 807, 5310, 121536, 2775630, 48782385, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Mar 25 2002

			n=4, 4!=24, A067734(24)=17 and the 17 solutions are as follows: {38,46,64,83,226,234,243,262,324,342,423,432,622,2223,2232,2322,3222}.

Cf. A067734, A000142, A068183, A068185.

a(n) = A067734(n!) = A067734(A000142(n)).

a(n) = 0 if n>10 since A068184 and A067185 are complete sequences.

More terms from Max Alekseyev, Sep 19 2009

cp's OEIS Frontend

A068191 Numbers n such that A067734(n)=0; complement of A002473; at least one prime-factor of n is larger than 7, it has 2 decimal digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A068189 Smallest positive number whose product of digits equals n, or a(n)=0 if no such number exists, i.e. when n has a prime divisor greater than 7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A068183 Largest number without decimal digits equal to 1 whose product of digits gives n!.

Views

Author

Keywords

Comments

Crossrefs

Formula

A068187 a(n) is the smallest number such that the product of its decimal digits equals n^n, or 0 if no solutions exist.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Extensions

A068184 Smallest number whose product of digits equals n!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A068185 Number of ways writing n^n as a product of decimal digits of some other number which has no digits equal to 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A068186 a(n) is the largest number whose product of decimal digits equals n^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A068190 Largest number whose digit product equals n; a(n)=0 if no such number exists, e.g., when n has a prime factor larger than 7; no digit=1 is permitted to avoid an infinite number of solutions.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula