A369184 a(n) is the first positive number that has exactly n anagrams which have 3 prime divisors, counted by multiplicity, or 0 if there is no such number.

1, 8, 103, 117, 156, 268, 1038, 1027, 1059, 1246, 1245, 1347, 1578, 3789, 10136, 10126, 10234, 10355, 10157, 10236, 10158, 11456, 10247, 10245, 10289, 10237, 10235, 10347, 10256, 10257, 10246, 10789, 10239, 10579, 12567, 10578, 13457, 12369, 14559, 12458, 12579, 23789, 24789, 12459, 100258, 12345

Offset

0,2

Comments

An anagram of a number k is a permutation of the base-10 digits of k with no leading 0's.

a(n) is the first k such that A175854(k) = n, or 0 if n is not a term of A175854.

Conjecture: all a(n) > 0.

Links

Robert Israel, Table of n, a(n) for n = 0..1000

Example

a(5) = 268 because 268 has 5 anagrams that have 3 prime divisors, counted by multiplicity, and is the first number that does that:
268 = 2^2 * 67, 286 = 2 * 11 * 13, 628 = 2^2 * 157, 682 = 2 * 11 * 31, 826 = 2 * 7 * 59.

Maple

f:= proc(n) local L, d, w, x, i;
      L:= convert(n, base, 10); d:= nops(L);
      L:= select(t -> t[-1] <> 0, combinat:-permute(L));
      L:= map(t-> add(t[i]*10^(i-1), i=1..d), L);
      nops(select(t -> numtheory:-bigomega(t)=3, L))
end proc:
g:= proc(xin, d, n) # first anagrams with n digits starting xin, all other digits >= d
  option remember;
  local i;
  if 1 + ilog10(xin) = n then return xin fi;
  seq(procname(10*xin+i, i, n), i=d..9)
end proc:
h:= proc(n) # first anagrams with n digits
  local i, j;
  seq(seq(g(i*10^j, i, n), j=n-1..0, -1), i=1..9)
end proc:
N:= 100: # for a(0) .. a(N)
V:= Array(0..N): count:= 0:
for i from 1 while count < N+1 do
  for x in [h(i)] while count < N+1 do
    v:= f(x);
    if v <= N and V[v] = 0 then V[v]:= x; count:= count+1; fi
  od
od:
convert(V, list);

Prog

(Python)
from sympy import primeomega
from sympy.utilities.iterables import multiset_permutations
from itertools import combinations_with_replacement, count, islice
def func(n): return sum(1 for p in multiset_permutations(str(n)) if p[0]!='0' and primeomega(int("".join(p)))==3)
def agen(): # generator of terms
    adict, n = dict(), 0
    for d in count(1):
        for f in "123456789":
            for r in combinations_with_replacement("0123456789", d-1):
                k = int(f+"".join(r))
                v = func(k)
                if v not in adict:
                    adict[v] = k
                    while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 44))) # Michael S. Branicky, Jan 15 2024

Crossrefs

Cf. A014612, A175854. All terms are in A179239.

Sequence in context: A305603 A372334 A333985 * A297069 A090237 A222664

Adjacent sequences: A369181 A369182 A369183 * A369185 A369186 A369187

Keyword

nonn,base

Author

Robert Israel, Jan 15 2024

Status

approved