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.

262713, 267338, 283628, 342713, 351678, 432713, 451676, 516469, 516657, 516675, 622713, 634838, 651674, 716655, 728364, 851673, 857297, 916465, 1262713, 1267338, 1283628, 1342713, 1351678, 1432713, 1451676, 1516469, 1516657, 1516675, 1622713, 1634838, 1651674

Offset: 1

Author

Bernard Schott, Jan 26 2021

Keywords

• nonn
• base

Comments

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.

Question: is it possible to get a longer string of integers with this rule?

From David A. Corneth, Feb 01 2021: (Start)

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)?

The least term k such that k + 6 is here is k = 56516718. Are there consecutive terms that differ by less than 6? (End)

Examples

			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.
		

References

• 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.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A327750.

Cf. A007954, A052382, A247888, A340908.

Programs

Mathematica

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 *)

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

Extensions

Terms from Amiram Eldar, Jan 26 2021