A364363 a(n) is the greatest number with n prime factors, counted with multiplicity, and no decimal digit occurring more than twice, or -1 if there is no such number.

1, 9988776655443322001, 9988776655443321002, 99887766554433120201, 99887766554433210201, 99887766554433221001, 99887766554433220101, 99887766554433221010, 99887766554433122010, 99887766554433220110, 99887766554433211020, 99887766554433212100, 99887766554433221100, 99887766554433200112

Offset: 0

Author

Zak Seidov and Robert Israel, Jul 20 2023

Keywords

• nonn
• base

Comments

a(n) = -1 for n sufficiently large, as there are only finitely many numbers with no digits occurring more than twice.

a(n) > 0 for n <= 29.

From Jon E. Schoenfield, Jul 25 2023: (Start)

a(n) = -1 for all n > 65, since no term has more than 20 digits; all 20-digit terms have digit sum 90 and thus are divisible by 9 and can have no more than log_3(9) + floor(log_2(99887766554433221100/9)) = 2 + 63 = 65 prime factors, counted with multiplicity (as 2^64 * 3^2 is a 21-digit number); and no term with fewer than 20 digits can have more than floor(log_2(9988776655443322100)) = 63 prime factors, counted with multiplicity.

a(n) > 0 for n = 0..54, 56, 57, and 62; for all other n, a(n) = -1. (End)

Examples

			a(3) = 99887766554433120201 = 3^2 * 11098640728270346689 has 3 prime factors with multiplicity and each digit 0 to 9 occurs twice, and is the largest such number.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 0..65

Crossrefs

Cf. A001222, A363963.

Programs

Maple

nextp:= proc(P) local k,m, newP,PL,PG,iv,i;
   m:= nops(P);
   for k from m-1 by -1 do
     if P[k] > P[k+1] then
       PL,PG:= selectremove(`<`,P[k+1..m],P[k]);
       iv:= max[index](PL);
       return [op(P[1..k-1]),PL[iv],op(sort([op(subsop(iv=P[k], PL)),op(PG)],`>`))]
      fi
   od
end proc:
V:= Array(0..21): V[0]:= 1:
P:= [seq(i$2,i=9..0,-1)]: count:= 1:
while count < 3 do
  x:= add(P[i]*10^(19-i),i=1..19):
  w:= numtheory:-bigomega(x);
  if w <= 2 and V[w] = 0 then V[w]:= x; count:= count+1; fi;
  P:= nextp(P);
od:
P:= [seq(i$2,i=9..0,-1)]:
while count < 22 do
  x:= add(P[i]*10^(20-i),i=1..20):
  w:= numtheory:-bigomega(x);
  if w <= 21 and V[w] = 0 then V[w]:= x; count:= count+1; fi;
  P:= nextp(P);
od:
convert(V,list);