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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.

%I A327750 #54 Sep 08 2022 08:46:24
%S A327750 28,128,214,239,266,318,326,364,494,497,563,598,613,637,695,819,1128,
%T A327750 1214,1239,1266,1318,1326,1364,1494,1497,1563,1598,1613,1637,1695,
%U A327750 1819,2114,2139,2168,2285,2313,2356,2369,2419,2594,2639,2791,3118,3126,3148,3213,3235
%N 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.
%C A327750 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.
%C A327750 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).
%C A327750 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
%C A327750 This subset is A247888. - _Bernard Schott_, Jul 22 2020
%H A327750 Michel Marcus, <a href="/A327750/b327750.txt">Table of n, a(n) for n = 1..10000</a>
%H A327750 Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, <a href="https://bookstore.ams.org/mcl-7/">Moscow Mathematical Olympiads, 2000-2005</a>, Level A, Problem 2, 2003; MSRI, 2011, p. 15 and 97.
%H A327750 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%e A327750 28 + 2*8 = 44 and 2*8 = 4*4 hence 28 is a term.
%e A327750 326 + 3*2*6 = 362 and 3*2*6 = 3*6*2 hence 326 is another term.
%t A327750 pd[n_] := Times @@ IntegerDigits[n]; aQ[n_] := (p = pd[n]) > 0 && pd[n + p] == p; Select[Range[5000], aQ] (* _Amiram Eldar_, Sep 24 2019 *)
%o A327750 (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
%o A327750 (PARI) isok(n) = my(d = digits(n), p); vecmin(d) && (p=vecprod(d)) && (vecprod(digits(n+p)) == p); \\ _Michel Marcus_, Sep 24 2019
%o A327750 (Python)
%o A327750 def test(n):
%o A327750     m, p = n, 1
%o A327750     while m > 0:
%o A327750         m, p = m//10, p*(m%10)
%o A327750     if p == 0:
%o A327750         return 0
%o A327750     m, q = n+p, 1
%o A327750     while m > 0:
%o A327750         m, q = m//10, q*(m%10)
%o A327750     return p == q
%o A327750 n, a = 0, 0
%o A327750 while n < 100:
%o A327750     a = a+1
%o A327750     if test(a):
%o A327750         n = n+1
%o A327750     print(n,a) # _A.H.M. Smeets_, Sep 25 2019
%Y A327750 Cf. A007954, A009994, A052382.
%Y A327750 Subsequences: A247888, A340907, A340908 (primitives).
%K A327750 nonn,base
%O A327750 1,1
%A A327750 _Bernard Schott_, Sep 24 2019
%E A327750 More terms from _Amiram Eldar_, Sep 24 2019