A239586 -id:A239586 - cp's OEIS frontend

A078972 Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits.

4, 6, 9, 10, 14, 15, 21, 25, 35, 49, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 341, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781

Offset: 1

Jason Earls, Jan 12 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." [Alpern]
Up to 10^8 the approximate sum of reciprocals is ~1.232884485... - Jason Earls, Oct 15 2010
Let f(n) = a(n) - floor(sqrt(a(n)))^2, or how much larger a(n) is than the largest square number <= a(n). Then f(n) is odd for all even squares, and even for all odd squares where n > 5. See "Ulam spiral" in links. - Christian N. K. Anderson, Jun 12 2013

			1711 = 29*59 is in the sequence since both of its factors have two digits.

P. D. James, The Private Patient, Knopf, NY, 2008, p. 192. (from N. J. A. Sloane, Aug 27 2009)

T. D. Noe, Table of n, a(n) for n = 1..10537
Dario Alpern, Brilliant Numbers.
Christian N. K. Anderson, Ulam Spiral of n=1..3000.

Cf. A001358, A085647, A084126, A084127, A055642, A085721, A376703, A376704.

Row 2 of A376738.

import Data.Function (on)
a078972 n = a078972_list !! (n-1)
a078972_list = filter brilliant a001358_list where
   brilliant x = (on (==) a055642) p (x `div` p) where p = a020639 x
-- Reinhard Zumkeller, Nov 10 2013, Mar 22 2014

fQ[n_] := Block[{fi = FactorInteger@n}, Plus @@ Last /@ fi == 2 && Floor[ Log[10, fi[[1, 1]] ]] == Floor[ Log[10, fi[[ -1, 1]] ]]]; Select[ Range@792, fQ@# &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[800],PrimeOmega[#]==2&&Length[Union[IntegerLength[FactorInteger[#][[;;,1]]]]]==1&] (* Harvey P. Dale, Jan 24 2025 *)
Select[Range@1000, Differences@IntegerLength@Flatten@(ConstantArray@@#&/@FactorInteger[#]) == {0} &] (* Hans Rudolf Widmer, Oct 25 2022 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors *)
Flatten[Array[dlist2, 2]] (* Paolo Xausa, Oct 05 2024 *)

is(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])) \\ Charles R Greathouse IV, Jun 16 2011

from sympy import sieve
A078972 = []
for n in range(3):
    pr = list(sieve.primerange(10**n,10**(n+1)))
    for i,p in enumerate(pr):
        for q in pr[i:]:
            A078972.append(p*q)
A078972 = sorted(A078972)
# Chai Wah Wu, Aug 26 2014

a(n) = A239585(n) * A239586(n). - Reinhard Zumkeller, Mar 22 2014

Edited by N. J. A. Sloane, Aug 27 2009

A239585 Prime factor <= other prime factor of n-th brilliant number, cf. A078972.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 5, 5, 7, 11, 11, 13, 11, 11, 13, 13, 11, 17, 13, 11, 17, 11, 19, 13, 17, 13, 11, 19, 11, 11, 13, 17, 11, 17, 23, 13, 19, 13, 11, 19, 13, 17, 11, 23, 11, 13, 17, 19, 23, 17, 11, 13, 19, 11, 13, 17, 11, 19, 29, 23, 11, 13, 19, 29, 17, 11

Offset: 1

Reinhard Zumkeller, Mar 22 2014

a(n) = A020639(A078972(n)) = A078972(n) / A239586(n).

A055642(a(n)) = A055642(A239586(n)).

			      n  | A239585(n) | A239586(n) | A078972(n)   Lengths of factors
  -------+------------+------------+-----------   ------------------
      1  |        2   |        2   |        4          1
      5  |        2   |        7   |       14
     10  |        7   |        7   |       49
         |.........................|              ..................
     11  |       11   |       11   |      121          2
     78  |       11   |       97   |     1067
    100  |       37   |       37   |     1369
    241  |       97   |       97   |     9409
         |.........................|              ..................
    242  |      101   |      101   |    10201          3
   1000  |      193   |      263   |    50759
   2530  |      101   |      997   |   100697
  10000  |      743   |      937   |   696191
  10537  |      997   |      997   |   994009
         |.........................|              ..................
  10538  |     1009   |     1009   |  1018081          4

Reinhard Zumkeller, Table of n, a(n) for n = 1..20000
Dario Alpern, Brilliant Numbers

Subsequence of A084126.

Cf. A020639, A055642, A078972, A239586.

Haskell
```
a239585 = a020639 . a078972
```

Table[With[{f = FactorInteger[k]}, If[Total[f[[All, 2]]] == 2 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1, f[[1, 1]], Nothing]], {k, 1000}] (* Paolo Xausa, Oct 02 2024 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Map[FactorInteger[#][[1, 1]]&, Flatten[Array[dlist2, 2]]] (* Paolo Xausa, Oct 09 2024 *)

cp's OEIS Frontend

A078972 Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits.

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