A254621 Zerofree numbers having product of digits less than or equal to sum of digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 31, 41, 51, 61, 71, 81, 91, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 131, 132, 141, 151, 161, 171, 181, 191, 211, 212, 213, 221, 231, 311, 312, 321, 411

Offset: 1

Author

David A. Corneth, Feb 03 2015

Keywords

• nonn
• base

Comments

Intersection of A052382 and A062996.

The repunit A002275(k), for k >= 2, appears at position A254622(k-1) + 1. - Wolfdieter Lang, Feb 23 2015

Links

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

Crossrefs

Cf. A052382, A062996, A062997, A062998, A062999, A254622, A002275.

Programs

Maple

extend:= proc(t, b, d)
  local i,j,m,s,p;
  p:= t[2];
  s:= t[3];
  if s = 0  then if b=2 then j:= 3 else j:= 2 fi
  else  for j from 0 to d-nops(t[1]) while p*b^j <= s + j*b do od
  fi:
  seq([[op(t[1]),b$i],p*b^i,s+i*b],i=0..j-1);
end proc:
f:= proc(d)
   local j, b, Res;
   Res:= [seq([[1$j],1,j],j=0..d)];
   for b from 2 to 9 do
     Res:= map(extend,Res,b,d)
   od:
   Res:= map(t -> op(combinat:-permute(t[1])),Res);
   subs(0=NULL,sort(map(t -> add(t[i]*10^(i-1),i=1..nops(t)), Res)));
end proc:
f(5); # Robert Israel, May 19 2015

Mathematica

m[w_] := Flatten@Table[i, {i,9}, {w[[i]]}]; a[upd_] := Union@ Flatten@ Table[ FromDigits /@ Flatten[Permutations /@ m /@ Select[ Flatten[ Permutations /@ (IntegerPartitions[d + 9, {9}, Range[d + 1]] - 1), 1], Times @@ (Range[9]^#) <= Total[# Range[9]] &], 1], {d,  upd}]; a[12] (* terms with up to 12 digits, Giovanni Resta, May 19 2015 *)
zfnQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&Times@@idn <= Total[ idn]]; Select[Range[500],zfnQ] (* Harvey P. Dale, Jun 29 2019 *)

PARI

is(n)={my(d=digits(n));my(p=prod(i=1,#d,d[i])); 0 < p && p<=vecsum(d) } \\ David A. Corneth, May 15 2015