OFFSET
0,2
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)
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 0..65
EXAMPLE
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.
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);
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Zak Seidov and Robert Israel, Jul 20 2023
STATUS
approved