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.

0, 1, 5, 6, 7, 11, 19, 79, 84, 111, 123, 176, 232, 396, 1111, 11111, 111111, 331788, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..27.

Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019.

Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019. [Cached copy, pdf file, with permission]

Example

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.
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.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A328095, A326806, A328544.

Sequence in context: A139205 A374960 A288859 * A039589 A373854 A028318

Adjacent sequences: A307748 A307749 A307750 * A307752 A307753 A307754

Keyword

nonn,base,more

Author

Scott R. Shannon, Nov 10 2019

Extensions

a(24)-a(27) from Giovanni Resta, Nov 15 2019

Status

approved