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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

"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

REFERENCES

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

LINKS

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.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(PARI) 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

(Haskell)

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

(Python)

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):