A369203 a(n) is the first number that has exactly n anagrams that each have exactly n prime divisors, counted by multiplicity.

2, 15, 117, 135, 1224, 10023, 10026, 50688, 104445, 100368, 1012257, 1002258, 1034568, 10027899, 10024569, 100002789, 100234566, 100236789, 1000024569, 1012566789, 10000224468, 10002367899, 10002345678, 100012344588, 100012234689, 100223456778, 1000012457889, 1002345566778

Offset

1,1

Comments

a(n) is the first number that has n anagrams k such that A001222(k) = n.

Does 9 divide a(n) for n > 6? - David A. Corneth, Jan 16 2024

Links

David A. Corneth, Table of n, a(n) for n = 1..61

Robert Israel, Anagrams of a(n) with n prime divisors, for n = 1 to 18

Example

a(4) = 135 is a term because 135 has 4 anagrams having 4 prime divisors, counted by multiplicity: 135 = 3^3 * 5, 315 = 3^2 * 5 * 7, 351 = 3^3 * 13 and 513 == 3^3 * 19, and no number < 135 works.
a(6) != 2367 because 2367 has exactly 7 anagrams with each having exactly 6 prime divisors (namely 2673, 3276, 3726, 6237, 6372, 6732, 7236). - David A. Corneth, Jan 16 2024

Maple

f:= proc(n) # numbers k such that n has k anagrams with Omega = k
        local L, W, WS, V, 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);
      W:= map(t -> numtheory:-bigomega(t), L);
      WS:= convert(W, set);
      for x in WS do V[x]:= 0 od;
      for x in W do V[x]:= V[x]+1 od;
      select(x -> V[x] = x, WS);
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:
V:= 'V': m:= 0:
for d from 1 to 9 do
  for x in h(d) do
    for y in f(x) do
      if not assigned(V[y]) then V[y]:= x: m:= max(m, y) fi
od od od:
seq(V[y], y=1..m);

Prog

(Python)
from collections import Counter
from sympy import primeomega as W
from sympy.utilities.iterables import multiset_permutations as MP
from itertools import combinations_with_replacement, count, islice
def counteq(n):
    c = Counter(W(int("".join(p))) for p in MP(str(n)) if p[0]!='0')
    return [i for i in c if c[i] == i]
def agen(): # generator of terms
    adict, n = dict(), 1
    for d in count(len(str(2**n))):
        for f in "123456789":
            for r in combinations_with_replacement("0123456789", d-1):
                k = int(f+"".join(r))
                for v in counteq(k):
                    if v not in adict:
                        adict[v] = k
                while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 8))) # Michael S. Branicky, Jan 16 2024

Crossrefs

Cf. A001222, A369184. All terms are in A179239.

Sequence in context: A052861 A300910 A161937 * A074621 A341929 A185758

Adjacent sequences: A369200 A369201 A369202 * A369204 A369205 A369206

Keyword

nonn,base

Author

Robert Israel, Jan 15 2024

Extensions

More terms from David A. Corneth, Jan 16 2024

Status

approved