A376703 3-brilliant numbers: numbers which are the product of three primes having the same number of decimal digits.

8, 12, 18, 20, 27, 28, 30, 42, 45, 50, 63, 70, 75, 98, 105, 125, 147, 175, 245, 343, 1331, 1573, 1859, 2057, 2197, 2299, 2431, 2717, 2783, 2873, 3179, 3211, 3289, 3509, 3553, 3751, 3757, 3887, 3971, 4147, 4199, 4301, 4433, 4477, 4693, 4807, 4901, 4913, 4961, 5083

Offset: 1

Author

Paolo Xausa, Oct 02 2024

Keywords

• nonn
• base

Examples

			4961 is a term because 4961 = 11 * 11 * 41, and these three prime factors have the same number of digits.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Dario Alpern, 3-Brilliant Numbers.

Crossrefs

Subsequence of A014612.

Cf. A078972, A083128, A083182, A376704.

Programs

Mathematica

A376703Q[k_] := With[{f = FactorInteger[k]}, Total[f[[All, 2]]] == 3 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1];
Select[Range[6000], A376703Q] (* or *)
dlist3[d_] := Sort[Times @@@ DeleteDuplicates[Map[Sort, Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 3]]]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Flatten[Array[dlist3, 2]]

Python

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) == 3 and len(set([len(str(p)) for p in f])) == 1
print([k for k in range(5100) if ok(k)]) # Michael S. Branicky, Oct 05 2024

Python

from math import prod
from sympy import primerange
from itertools import count, combinations_with_replacement as cwr, islice
def bgen(d): # generator of terms that are products of d-digit primes
    primes, out = list(primerange(10**(d-1), 10**d)), set()
    for t in cwr(primes, 3): out.add(prod(t))
    yield from sorted(out)
def agen(): # generator of terms
    for d in count(1): yield from bgen(d)
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 05 2024