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A307751 Numbers k such that the number m multiplied by the product of all its digits contains k as a substring, where m = k multiplied by the product of all its digits.

%I A307751 #37 Dec 21 2024 13:46:57
%S A307751 0,1,5,6,7,11,19,79,84,111,123,176,232,396,1111,11111,111111,331788,
%T A307751 1111111,11111111,111111111,1111111111,11111111111,111111111111,
%U A307751 1111111111111,11111111111111,111111111111111
%N A307751 Numbers k such that the number m multiplied by the product of all its digits contains k as a substring, where m = k multiplied by the product of all its digits.
%C A307751 Inspired by A328095. Like A328095 this sequence contains all the repunits. These numbers could be called 'Two-step Revenant numbers'. It is unknown if 331788 is the last non-repunit.
%H A307751 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2019/10/revenant-numbers.html">Revenant Numbers</a>, Cinquante Signes, Oct 19 2019.
%H A307751 Eric Angelini, <a href="/A328095/a328095.pdf">Revenant Numbers</a>, Cinquante Signes, Oct 19 2019. [Cached copy, pdf file, with permission]
%e A307751 79 is in the sequence as m = 79*7*9 = 4977, and 4977*4*9*7*7 = 8779428, and '8779428' contains '79' as a substring.
%e A307751 331788 is in the sequence as m = 331788*3*3*1*7*8*8 = 1337769216, and 1337769216*1*3*3*7*7*6*9*2*1*6 = 382291633317888, and '382291633317888' contains '331788' as a substring.
%t A307751 f[n_] := n * Times @@ IntegerDigits[n]; aQ[n_] := SequenceCount[IntegerDigits[ f[f[n]] ], IntegerDigits[n]] > 0; Select[Range[0, 10^6], aQ] (* _Amiram Eldar_, Nov 10 2019 *)
%o A307751 (Magma) a:=[0]; f:=func<n|n*(&*Intseq(n))>; for k in [1..1200000] do t:=IntegerToString(f(f(k))); s:=IntegerToString(k); if s in t then Append(~a,k); end if; end for; a; // _Marius A. Burtea_, Nov 10 2019
%Y A307751 Cf. A328095, A326806, A328544.
%K A307751 nonn,base,more
%O A307751 1,3
%A A307751 _Scott R. Shannon_, Nov 10 2019
%E A307751 a(24)-a(27) from _Giovanni Resta_, Nov 15 2019