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

1,1

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 *)
brilQ[n_]:=Module[{fin=FactorInteger[n]}, Total[Transpose[fin][[2]]]==2 &&  Length[Union[IntegerLength[Transpose[fin][[1]]]]]==1]; Select[Range[1000], brilQ]  (* Harvey P. Dale, Feb 06 2011 *)
Select[Range@1000, Differences@IntegerLength@Flatten@(ConstantArray@@#&/@FactorInteger[#]) == {0} &] (* Hans Rudolf Widmer, Oct 25 2022 *)

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):
        for q in pr[i:]:
            A078972.append(p*q)
A078972 = sorted(A078972)
# Chai Wah Wu, Aug 26 2014

Crossrefs

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

Sequence in context: A115654 A362910 A036326 * A115652 A337372 A317299

Adjacent sequences: A078969 A078970 A078971 * A078973 A078974 A078975

Keyword

base,easy,nonn

Author

Jason Earls, Jan 12 2003

Extensions

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

Status

approved