A376063 a(n) is the least number that ends with exactly n 1's and has exactly n prime divisors, counted with multiplicity.

31, 411, 3111, 501111, 2211111, 129111111, 291111111, 7311111111, 882111111111, 23481111111111, 74511111111111, 8385111111111111, 419541111111111111, 15531711111111111111, 169059111111111111111, 641451111111111111111, 457680711111111111111111, 138816111111111111111111

Offset

1,1

Comments

a(n) exists for all n: by Dirichlet's theorem on primes in arithmetic progressions, there is a prime x such that 3^(n-1) * x == 2*10^n + (10^n-1)/9 mod 10^(n+1).

Links

Table of n, a(n) for n=1..18.

Example

a(4) = 501111 because 501111 = 3^2 * 13 * 4283 is the product of four primes (counted with multiplicity) and ends in four 1's, and no smaller number works.

Maple

f:= proc(n) local x, t, y;
  t:= (10^n-1)/9;
  for x from 0 do
    if x mod 10 = 1 then next fi;
    y:= 10^n * x + t;
    if numtheory:-bigomega(y) = n then return y fi
  od;
end proc:
map(f, [$1..14]);

Prog

(Python)
from sympy import factorint
from itertools import count
def f(n): return sum(e for p, e in factorint(n).items())
def a(n):
    Rn, pow10 = (10**n-1)//9, 10**n
    return next(t for d in count(0) for r in range(10**(d-1) if d else 0, 10**d) if r%10 != 1 and f(t:=r*pow10 + Rn) == n)
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Sep 21 2025

Crossrefs

Sequence in context: A297019 A022691 A125443 * A268137 A089375 A069582

Adjacent sequences: A376060 A376061 A376062 * A376064 A376065 A376066

Keyword

nonn,base

Author

Robert Israel, Sep 07 2024

Status

approved