A386761 a(n) is the least number with exactly two distinct base-10 digits and exactly n distinct prime factors, or -1 if there is no such number.

13, 10, 30, 330, 6006, 30030, 1111110, 200222022, 1001110110, 955559599995, 1000110001000110, 11011101101110110, 11111100000101111010, 200220200202022022022

Offset

1,1

Comments

a(n) is the least term k of A031955, if it exists, such that A001221(k) = n.

Links

Table of n, a(n) for n=1..14.

Example

a(4) = 330 because 330 has two distinct digits 3 and 0, and four distinct prime factors 2, 3, 5 and 11, and no smaller number works.

Maple

G:= proc(d)
  local S, a, b, s, q, r, i;
  S:= {}:
  for s in combinat:-powerset({$0..d-1}) minus {{}, {$0..d-1}} do
    q:= add(10^i, i=s);
    r:= (10^d-1)/9 - q;
    S:= S union {seq(seq(a*q+b*r, b = a+1..9), a = `if`(q < 10^(d-1), 0, 1) .. 8)};
 od;
S:= sort(convert(S, list));
end proc:
M:= 12: # for a(1) .. a(M)
V:= Vector(M): count:= 0:
for d from 2 while count < M do
for x in G(d) while count < M do
    v:=NumberTheory:-NumberOfPrimeFactors(x, distinct);
    if v <= M and V[v] = 0 then
       count:= count+1; V[v]:= x;
    fi
  od
od:
convert(V, list);

Prog

(Python)
from sympy import factorint
from itertools import count, product, islice
def bgen(): # generator of A031955
    for d in count(2):
        yield from sorted(int("".join(t)) for f in "123456789" for r in "0123456789" if f!=r for s in product(f+r, repeat=d-1) if len(set(t:=(f, )+s)) == 2)
def agen(): # generator of terms
    n, adict = 1, dict()
    for t in bgen():
        v = len(factorint(t))
        if v not in adict:
            adict[v] = t
            while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 9))) # Michael S. Branicky, Oct 11 2025

Crossrefs

Cf. A001221, A031955.

Sequence in context: A291425 A180864 A206608 * A094813 A364712 A152298

Adjacent sequences: A386758 A386759 A386760 * A386762 A386763 A386764

Keyword

nonn,base,more

Author

Robert Israel, Oct 06 2025

Extensions

a(13) from Michael S. Branicky, Oct 11 2025

a(14) from Daniel Suteu, Oct 22 2025

Status

approved