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

Author

Erich Friedman

Keywords

• nonn
• base
• nice
• easy

Comments

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

Examples

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

Links

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

Crossrefs

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

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

Programs

Haskell

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

Magma

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

Mathematica

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

PARI

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

Extensions

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