A249517 Numbers k for which the digital sum A007953(k) and the digital product A007954(k) both contain the same distinct digits as the number k.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11111111111

Offset: 1

Author

Jaroslav Krizek, Oct 31 2014

Keywords

• nonn
• base

Comments

a(12) = (10^106-1)/9 + 122222222. - Max Alekseyev, Nov 15 2014

Other entries include (10^111-1)/9, (10^113-1)/9 + 177, (10^115-1)/9 + 122222222, (10^117-1)/9 + 11117, (10^125-1)/9 + 2224, (10^126-1)/9 + 333335, (10^135-1)/9 + 4666, (10^143-1)/9 + 446, (10^143-1)/9 + 2224, (10^144-1)/9 + 33335. All other entries with 150 or fewer digits are formed by permutations of the decimal digits of these entries (including a(12)). (10^((10^m-1)/9)-1)/9 are entries of the sequence for m >= 0. - Chai Wah Wu, Nov 15 2014

Examples

			11111111111 is a term since A007953(11111111111) = 11 and A007954(11111111111) = 1.
		

Crossrefs

Intersection of A249515 and A249516. Subsequence of A249334.

Cf. A007954, A249334, A249515, A249516.

Programs

Magma

[0] cat [n: n in [0..10^7] | Set(Intseq(n)) eq Set(Intseq(&*Intseq(n))) and Set(Intseq(n)) eq Set(Intseq(&+Intseq(n)))];

PARI

is(n)=if(n<=9,return(1)); my(d=digits(n),s=Set(d)); s==Set(digits(sum(i=1,#d,d[i]))) && s==Set(digits(prod(i=1,#d,d[i]))) \\ Charles R Greathouse IV, Nov 13 2014

Python

from itertools import product
from operator import mul
from functools import reduce
A249517_list = [0]
for g in range(1,15):
    xp, ylist = [], []
    for i in range(9*g,-1,-1):
        x = set(str(i))
        if not (('0' in x) or (x in xp)):
            xv = [int(d) for d in x]
            imin = int(''.join(sorted(str(i))))
            if max(xv)*(g-len(x)) >= imin-sum(xv) and i-sum(xv) >=  min(xv)*(g-len(x)):
                xp.append(x)
                for y in product(x,repeat=g):
                    if set(y) == x:
                        yd = [int(d) for d in y]
                        if set(str(sum(yd))) == x == set(str(reduce(mul, yd, 1))):
                            ylist.append(int(''.join(y)))
    A249517_list.extend(sorted(ylist)) # Chai Wah Wu, Nov 15 2014

Extensions

a(11) = 11111111111 confirmed by Sean A. Irvine, Nov 13 2014, by direct search.