A083128 -id:A083128 - cp's OEIS frontend

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

Paolo Xausa, Oct 02 2024

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

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

Subsequence of A014612.

Cf. A078972, A083128, A083182, A376704.

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]]

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

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

A083182 Greatest 3-brilliant number of size n.

Original entry on oeis.org

8, 98, 343, 9971, 99937, 912673, 9999707, 99999667, 991026973, 9999999467, 99999999007, 991921850317, 9999999994771, 99999999994117, 999730024299271, 9999999999997097, 99999999999992023, 999949000866995087

Offset: 1

Robert G. Wilson v, May 11 2003

Brilliant numbers, as defined by Peter Wallrodt, are numbers with two prime factors of the same length (in decimal notation). These numbers are generally used for cryptographic purposes and for testing the performance of prime factoring programs.

a(3n) will always be the cube of the greatest prime less than 10^n.

			a(5) = 99937 = 37 * 37 * 73 and there is no greater number of five digits which has three prime factors, not necessarily different, of the same size in decimal notation.

Dario Alpern, Brilliant numbers

Cf. A083128.

A376705 a(n) = smallest 4-brilliant number (A376704) with n decimal digits.

Original entry on oeis.org

16, 100, 1029, 14641, 100529, 1000109, 10005647, 104060401, 1000002847, 10000000861, 100000024219, 1036488922561, 10000000051429, 100000000009729, 1000000000011799, 10028029413722401, 100000000000035241, 1000000000000012861, 10000000000000080101, 100012000540010800081

Offset: 2

Paolo Xausa, Oct 02 2024

See Alpern link for more terms, along with their prime factorization.

			a(6) = 100529 because 100529 = 11 * 13 * 19 * 37 is the smallest 6-digit number with four prime factors having the same number of digits.

Dario Alpern, 4-Brilliant Numbers.

Subsequence of A376704.

Cf. A078972, A083128, A376703, A376706.

A347818 Smallest n-digit brilliant number.

Original entry on oeis.org

4, 10, 121, 1003, 10201, 100013, 1018081, 10000043, 100140049, 1000000081, 10000600009, 100000000147, 1000006000009, 10000000000073, 100000380000361, 1000000000000003, 10000001400000049, 100000000000000831, 1000000014000000049, 10000000000000000049, 100000000380000000361

Offset: 1

Eric Chen, Sep 15 2021

A brilliant number is a semiprime (products of two primes, A001358) whose two prime factors have the same number of decimal digits. For an n-digit brilliant number, the two prime factors must each have ceiling(n/2) decimal digits.

Since all brilliant numbers are semiprimes, a(n) >= A098449(n), also, a(n) = A098449(n) for n = 1, 2, 4, 16, 78, ..., are there infinitely many n such that a(n) = A098449(n)?

			a(6) =    100013 =   103 * 971.
a(7) =   1018081 =  1009 * 1009.
a(8) =  10000043 =  2089 * 4787.
a(9) = 100140049 = 10007 * 10007.

Dario Alejandro Alpern, Brilliant numbers
World of Numbers, Smallest n-digit prp

Cf. A078972, A003617, A083289, A084475, A084476, A083128, A098449.

Join[{4,10},Table[Module[{k=1},While[PrimeOmega[10^n+k]!=2||Length[ Union[ IntegerLength/@ FactorInteger[ 10^n+k][[;;,1]]]]!=1,k+=2];10^n+k],{n,2,20}]] (* Harvey P. Dale, Jan 09 2024 *)

isA078972(n)=my(f=factor(n)); (#f[, 1]==1 && f[1, 2]==2) || (#f[, 1]==2 && f[1, 2]==1 && f[2, 2]==1 && #Str(f[1, 1])==#Str(f[2, 1]))
A084476(n)=for(k=0,10^n,if(isA078972(10^(2*n-1)+k),return(k)))
a(n)=if(n%2,nextprime(10^((n-1)/2))^2,10^(n-1)+A084476(n/2)) \\ after Charles R Greathouse IV in A078972

a(n) = 10^(n-1) + A083289(n).
a(2*n) = 10^(2*n-1) + A084476(n).
a(2*n+1) = A003617(n+1)^2.
a(n) >= A098449(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

A083182 Greatest 3-brilliant number of size n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376705 a(n) = smallest 4-brilliant number (A376704) with n decimal digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A347818 Smallest n-digit brilliant number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula