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
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.
Programs
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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]]
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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
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