A340271 Positive integers with no digit equal to 0 or 1 such that when one iterates x -> A340270(x) starting with n, one eventually enters a cycle containing an integer greater than 9.

25, 26, 28, 34, 35, 36, 38, 43, 46, 52, 53, 62, 63, 64, 82, 83, 236, 239, 246, 254, 296, 326, 329, 362, 392, 426, 462, 524, 542, 926, 962

Offset

1,1

Comments

This was suggested by postings by Eric Angelini and Allan Wechsler to Math-Fun Mailing List, Dec. 29, 2020. It is an open question whether or not this sequence is complete. In fact, it is open whether or not the sequence of iterates of x -> A340270(x) starting with n always enters a cycle. There are no counterexamples up to 10^8.

As Allan Wechsler pointed out, if x contains 0 then A340270(x) = 1 and if x contains a digit 1 then A340270(x) is x with the 1 deleted.

From M. F. Hasler, Jan 03 2021: (Start)

The trajectory of all these integers ends in one of 3 cycles:

C(25) = (25) (also for n = 52, for n = 35 and 53 via 125, and for n = 38 and 83 via 512, and for n = 239, 329 and 392 via 1521 and 152);

C(26) = (26, 36, 216) (also for n = 62 and 63, for n = 246 and 426 and 462 via 2116 and 216);

C(926) = (926, 9216): also for 296 and 962, for 28, 34, 43 and 82 via 64 and 1296, also for 46, 236, 326 and 362 via 1296, and for 254, 524 and 54 via 2916. (End)

Links

Table of n, a(n) for n=1..31.

Example

Iterating the function x->A340270(x) starting with 34 we get 34 --> 4^3 = 64 --> 6^4 = 1296 --> 296^1 = 296 --> 96^2 = 9216 --> 926^1 = 926 --> 96^2 = 9216. So 34 is in this sequence.

Maple

leastdigit:=proc(n) min(convert(n, base, 10)); end:
locationdigit:=proc(n, d) local L, i;
  L:=convert(n, base, 10);
  for i from 1 to nops(L) do
   if d = L[i] then return (nops(L)+1-i); fi;
  od:
end:
cutout:=proc(X, i) [seq(X[j], j=1..i-1), seq(X[j], j=i+1..nops(X))]; end:
ToNum:=proc(X) add(X[i]*10^(nops(X)-i), i=1..nops(X)); end:
removeleastdigit:=proc(n) local i, X;
  i:=locationdigit(n, leastdigit(n));
  X:=ListTools:-Reverse(convert(n, base, 10));
  ToNum(cutout(X, i));
end proc:
h:=proc(n) removeleastdigit(n)^leastdigit(n); end:
F:=proc(N) local i, L;
  if h(N) = N then return [N]; fi;
  L:=[N];
  for i from 1 do
   L:=[op(L), h(L[-1])];
   if nops(L) > nops(convert(L, set)) then return L; fi;
  od:
  fail;
end proc:
G:=proc(X) if X[-1] < 10 then return "S"; else return "L"; fi; end proc:
L:=NULL:
for n from 1 to 1000 do
  sss:=convert(n, base, 10):
  if has(0, sss) or has(1, sss) then next; fi;
  u:=F(n);
  if u = fail then print(n, u); break; fi;
  if G(u) = "L" then L:=L, n; fi;
od:
L;

Prog

(Python)
def ok(n):
  strn = str(n)
  if "0" in strn or "1" in strn: return False
  iterates = {n}
  n = A340270(n)
  while n not in iterates:
    iterates.add(n)
    n = A340270(n)
  repeated = n
  if n > 9: return True
  n = A340270(n)
  while n != repeated:
    n = A340270(n)
    if n > 9: return True
  return False
print([n for n in range(1, 1001) if ok(n)]) # Michael S. Branicky, Jan 02 2021
(PARI) [n | n<-[1..9999], vecmin(digits(n))>1 && orbit(n)[1]>9] \\ with:
orbit(n, U=[n], m)={ while(n>9, my( m=vecmin(n=digits(n)));
  forstep( i=#n, 1, -1, if( n[i]==m, n=fromdigits(n[^i])^m; break)); setsearch(U, n) && break; U=setunion(U, [n])); U} \\ M. F. Hasler, Jan 03 2021

Crossrefs

Cf. A340270; A054054 (smallest digit of n).

Subset of A274126 (numbers with all digits larger than 1).

Sequence in context: A199992 A242221 A186538 * A003996 A132415 A292931

Adjacent sequences: A340268 A340269 A340270 * A340272 A340273 A340274

Keyword

base,nonn,more

Author

W. Edwin Clark, Jan 02 2021

Status

approved