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A078972
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Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits.
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83
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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)
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OFFSET
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1,1
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COMMENTS
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"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
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REFERENCES
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P. D. James, The Private Patient, Knopf, NY, 2008, p. 192. (from N. J. A. Sloane, Aug 27 2009)
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LINKS
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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.
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FORMULA
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a(n) = A239585(n) * A239586(n). - Reinhard Zumkeller, Mar 22 2014
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EXAMPLE
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1711 = 29*59 is in the sequence since both of its factors have two digits.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001358, A085647, A084126, A084127, A055642, A085721.
Sequence in context: A113433 A115654 A036326 * A115652 A337372 A317299
Adjacent sequences: A078969 A078970 A078971 * A078973 A078974 A078975
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KEYWORD
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base,easy,nonn
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AUTHOR
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Jason Earls, Jan 12 2003
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EXTENSIONS
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Edited by N. J. A. Sloane, Aug 27 2009
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STATUS
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approved
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