A068184 -id:A068184 - 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

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

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.

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

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

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Extensions