A327750 Numbers without zero digits such that after adding the product of its digits to it, a number with the same product of digits is obtained.

28, 128, 214, 239, 266, 318, 326, 364, 494, 497, 563, 598, 613, 637, 695, 819, 1128, 1214, 1239, 1266, 1318, 1326, 1364, 1494, 1497, 1563, 1598, 1613, 1637, 1695, 1819, 2114, 2139, 2168, 2285, 2313, 2356, 2369, 2419, 2594, 2639, 2791, 3118, 3126, 3148, 3213, 3235

Offset

1,1

Comments

The idea of this sequence comes from a problem in the annual Moscow Mathematical Olympiad (MMO) in 2003: Level A, problem 2. The problem only asks to find a ten-digit number that has the property of the name.

When an integer k belongs to this sequence, the integer 111..11//k obtained by concatenation // of 111..11 and k is also a term; hence, there are primitive terms as 28, 214, 239, 266, 318, 326, ... (A340908).

A subset of it is formed by the numbers 239, 326, 364, 497, 563, 598, 613, 637, 695, 819, 1239, 1326, 1364, 1497, 1563, 1598, 1613, 1637, 1695, 1819, 2139, 2313, 2356, 2369, ... for which the number obtained after adding the product of the digits has exactly the same digits (they are obtained by permuting the digits of the initial number). So, 239 + 2*3*9 = 239 + 54 = 293, 326 + 3*2*6 = 326 + 36 = 362, 3235 + 3*2*3*5 = 3235 + 90 = 3325, 23286 + 2*3*2*8*6 = 23286 + 576 = 23862. - Marius A. Burtea, Sep 24 2019

This subset is A247888. - Bernard Schott, Jul 22 2020

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, p. 15 and 97.

Index to sequences related to Olympiads.

Example

28 + 2*8 = 44 and 2*8 = 4*4 hence 28 is a term.
326 + 3*2*6 = 362 and 3*2*6 = 3*6*2 hence 326 is another term.

Mathematica

pd[n_] := Times @@ IntegerDigits[n]; aQ[n_] := (p = pd[n]) > 0 && pd[n + p] == p; Select[Range[5000], aQ] (* Amiram Eldar, Sep 24 2019 *)

Prog

(Magma) [k:k in [1..3500]| not 0 in Intseq(k) and &*Intseq(k) eq &*(Intseq(k+&*Intseq(k)))]; // Marius A. Burtea, Sep 24 2019
(PARI) isok(n) = my(d = digits(n), p); vecmin(d) && (p=vecprod(d)) && (vecprod(digits(n+p)) == p); \\ Michel Marcus, Sep 24 2019
(Python)
def test(n):
    m, p = n, 1
    while m > 0:
        m, p = m//10, p*(m%10)
    if p == 0:
        return 0
    m, q = n+p, 1
    while m > 0:
        m, q = m//10, q*(m%10)
    return p == q
n, a = 0, 0
while n < 100:
    a = a+1
    if test(a):
        n = n+1
    print(n, a) # A.H.M. Smeets, Sep 25 2019

Crossrefs

Cf. A007954, A009994, A052382.

Subsequences: A247888, A340907, A340908 (primitives).

Sequence in context: A339992 A320885 A045823 * A044360 A044741 A318778

Adjacent sequences: A327747 A327748 A327749 * A327751 A327752 A327753

Keyword

nonn,base

Author

Bernard Schott, Sep 24 2019

Extensions

More terms from Amiram Eldar, Sep 24 2019

Status

approved