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A340907 Numbers m without zero digits such that pod(q) = pod(k) = pod(m) where q = k + pod(k) and k = m + pod(m) where pod is the product of digits, A007954.

%I A340907 #43 Feb 01 2021 19:40:12
%S A340907 262713,267338,283628,342713,351678,432713,451676,516469,516657,
%T A340907 516675,622713,634838,651674,716655,728364,851673,857297,916465,
%U A340907 1262713,1267338,1283628,1342713,1351678,1432713,1451676,1516469,1516657,1516675,1622713,1634838,1651674
%N A340907 Numbers m without zero digits such that pod(q) = pod(k) = pod(m) where q = k + pod(k) and k = m + pod(m) where pod is the product of digits, A007954.
%C A340907 The idea of this sequence comes from a remark of _Amiram Eldar_ in Discussion section of A327750 (m + pod(m) = k with pod(k) = pod(m)) in September 2019.
%C A340907 Question: is it possible to get a longer string of integers with this rule?
%C A340907 From _David A. Corneth_, Feb 01 2021: (Start)
%C A340907 The product of digits of a(n) is a multiple of 6. In terms below 10^10 however all products of digits of a(n) are a multiple of 36. Is that product a multiple of 36 for all a(n)?
%C A340907 The least term k such that k + 6 is here is k = 56516718. Are there consecutive terms that differ by less than 6? (End)
%D A340907 Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, and Ivan Yashchenko, Moscow-Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, p. 15 and 97.
%H A340907 David A. Corneth, <a href="/A340907/b340907.txt">Table of n, a(n) for n = 1..10000</a>
%e A340907 262713 + pod(262713) = 262713 + 504 = 263217, whose product of digits is also 504, and 263217 + 504 = 263721 whose product of digits is again 504; hence, m=262713, k=263217, q=263721 and pod(m)=pod(k)=pod(q)=504, so 262713 is a term.
%t A340907 pod[n_] := Times @@ IntegerDigits[n]; seqQ[n_] := Module[{p = pod[n], k, q}, k = n + p; q = k + pod[k]; p > 0 && Equal @@ {p, pod[k], pod[q]}]; Select[Range[2*10^6], seqQ] (* _Amiram Eldar_, Jan 26 2021 *)
%o A340907 (PARI) isok(m) = my(pm=vecprod(digits(m)), k=m+pm, pk=vecprod(digits(k)), q=k+pk, pq=vecprod(digits(q))); pm && (pm==pk) && (pk==pq); \\ _Michel Marcus_, Jan 26 2021
%Y A340907 Subsequence of A327750.
%Y A340907 Cf. A007954, A052382, A247888, A340908.
%K A340907 nonn,base
%O A340907 1,1
%A A340907 _Bernard Schott_, Jan 26 2021
%E A340907 Terms from _Amiram Eldar_, Jan 26 2021