A249334 -id:A249334 - cp's OEIS frontend

A034710 Positive numbers for which the sum of digits equals the product of digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 132, 213, 231, 312, 321, 1124, 1142, 1214, 1241, 1412, 1421, 2114, 2141, 2411, 4112, 4121, 4211, 11125, 11133, 11152, 11215, 11222, 11251, 11313, 11331, 11512, 11521, 12115, 12122, 12151, 12212, 12221, 12511

Offset: 1

Erich Friedman

Positive numbers k such that A007953(k) = A007954(k).
If k is a term, the digits of k are solutions of the equation x1*x2*...*xr = x1 + x2 + ... + xr; xi are from [1..9]. Permutations of digits (x1,...,xr) are different numbers k with the same property A007953(k) = A007954(k). For example: x1*x2 = x1 + x2; this equation has only 1 solution, (2,2), which gives the number 22. x1*x2*x3 = x1 + x2 + x3 has a solution (1,2,3), so the numbers 123, 132, 213, 231, 312, 321 have the property. - Ctibor O. Zizka, Mar 04 2008
Subsequence of A249334 (numbers for which the digital sum contains the same distinct digits as the digital product). With {0}, complement of A249335 with respect to A249334. Sequence of corresponding values of A007953(a(n)) = A007954(a(n)): 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ... contains only numbers from A002473. See A248794. - Jaroslav Krizek, Oct 25 2014
There are terms of the sequence ending in any term of A052382. - Robert Israel, Nov 02 2014
The number of digits which are not 1 in a(n) is O(log log a(n)) and tends to infinity as a(n) does. - Robert Dougherty-Bliss, Jun 23 2020

			1124 is a term since 1 + 1 + 2 + 4 = 1*1*2*4 = 8.

Alois P. Heinz, Table of n, a(n) for n = 1..27597 (first 1200 terms from T. D. Noe)

Cf. A002473, A007953, A007954, A052382, A061672, A248794, A249334, A249335.

Cf. A066306 (prime terms), A066307 (nonprimes).

import Data.List (elemIndices)
a034710 n = a034710_list !! (n-1)
a034710_list = elemIndices 0 $ map (\x -> a007953 x - a007954 x) [1..]
-- Reinhard Zumkeller, Mar 19 2011

[n: n in [1..10^6] | &*Intseq(n) eq &+Intseq(n)] // Jaroslav Krizek, Oct 25 2014

Select[Range[12512], (Plus @@ IntegerDigits[ # ]) == (Times @@ IntegerDigits[ # ]) &] (* Alonso del Arte, May 16 2005 *)

is(n)=my(d=digits(n)); vecsum(d)==factorback(d) \\ Charles R Greathouse IV, Feb 06 2017

Corrected by Larry Reeves (larryr(AT)acm.org), Jun 27 2001

Definition changed by N. J. A. Sloane to specifically exclude 0, Sep 22 2007

A061672 Smallest positive number formed by a set of digits whose product = sum of the digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 1124, 11125, 11133, 11222, 111126, 1111127, 1111134, 11111128, 11111223, 111111129, 111111135, 1111111144, 11111111136, 11111111224, 111111112222, 1111111111137, 1111111111145, 1111111111233

Offset: 1

Amarnath Murthy, Jun 26 2001

From M. F. Hasler, Oct 29 2014: (Start)
This is the subsequence of terms of A034710 with digits in nondecreasing order, which is meant by "smallest": For example, 132 also has sum of digits = product of digits, but is already "represented" by 123. The word "set" in the definition actually means "multiset".
The sequence is infinite: for any number N whose digits form a nondecreasing sequence whose sum of digits S is not larger than the product of digits P (i.e., N in A062998), a term of the sequence is obtained by prefixing N with P-S digits '1'. (End)

			1124 is a term since 1 + 1 + 2 + 4 = 1*1*2*4 = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..667 (terms n=1..134 from T. D. Noe).
Sean A. Irvine, Java program (github)

Cf. A034710, A249334.

is_A061672(n)={vecsort(n=digits(n))==n && normlp(n,1)==prod(i=1,#n,n[i])} \\ M. F. Hasler, Oct 29 2014

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 27 2001
Corrected by Franklin T. Adams-Watters, Oct 25 2006
Further corrections from T. D. Noe, Oct 12 2007

A249335 Numbers n for which the digital sum contains the same distinct digits as the digital product but the digital sum is not equal to the digital product.

Original entry on oeis.org

99, 1137, 1173, 1317, 1371, 1713, 1731, 3117, 3171, 3344, 3434, 3443, 3711, 4334, 4343, 4433, 7113, 7131, 7311, 11558, 11585, 11855, 15158, 15185, 15518, 15581, 15815, 15851, 18155, 18515, 18551, 22334, 22343, 22433, 23234, 23243, 23324, 23342, 23423, 23432

Offset: 1

Jaroslav Krizek, Oct 25 2014

Numbers n such that A007953(n) contains the same distinct digits as A007954(n) but A007953(n) is not equal to A007954(n).

Complement of A034710 with respect to A249334.

			1137 is a member since 1+1+3+7 = 12 and 1*1*3*7 = 21.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..100 from Jaroslav Krizek)

Cf. A034710, A007953, A007954, A249334.

[n: n in [1..10^6] | (&*Intseq(n)) ne (&+Intseq(n)) and Set(Intseq(&*Intseq(n))) eq Set(Intseq(&+Intseq(n)))]

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.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Oct 31 2014

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

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

Intersection of A249515 and A249516. Subsequence of A249334.

Cf. A007954, A249334, A249515, A249516.

[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)))];

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

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

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

A249443 Numbers with digits in nondecreasing order and digital sum not larger than the product of the digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 123, 124, 125, 126, 127, 128, 129, 133, 134, 135, 136, 137, 138, 139, 144, 145, 146, 147, 148, 149, 155, 156, 157, 158

Offset: 1

M. F. Hasler, Oct 29 2014

Intersection of A009994 and A062998.
Except for the initial 0, a subsequence of the zeroless numbers A052382.
The nonzero terms of this sequence correspond to a term of A061672 obtained by concatenation with A002275(A007954(a(n))-A007953(a(n))).

Cf. A007953, A007954, A034710, A061672, A249334.

is(n)={vecsort(n=digits(n))==n && normlp(n,1)<=prod(i=1,#n,n[i])}

A338257 Positive integers k with digits in nondecreasing order for which the digital sum contains the same distinct digits as the digital product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 99, 123, 1124, 1137, 3344, 11125, 11133, 11222, 11558, 22334, 111126, 111339, 222233, 1111127, 1111134, 1111278, 1112779, 11111128, 11111223, 11111478, 11111479, 11112455, 111111129, 111111135, 111111447, 111111559, 111111667, 111112278

Offset: 1

David A. Corneth, Oct 18 2020

Intersection of A009994 and A249334.

			3344 is in the sequence as its digits are in nondecreasing order and the digital sum is 14 and the digital product is 144. The digits of the latter two are either 1 or 4.

Cf. A009994, A249334.

is(n) = {my(d); if(vecsort(d = digits(n)) != d, return(0)); Set(digits(vecprod(d))) == Set(digits(vecsum(d)))}

cp's OEIS Frontend

A034710 Positive numbers for which the sum of digits equals the product of digits.

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A061672 Smallest positive number formed by a set of digits whose product = sum of the digits.

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A249335 Numbers n for which the digital sum contains the same distinct digits as the digital product but the digital sum is not equal to the digital product.

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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.

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A249443 Numbers with digits in nondecreasing order and digital sum not larger than the product of the digits.

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A338257 Positive integers k with digits in nondecreasing order for which the digital sum contains the same distinct digits as the digital product.

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PARI