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%I A078972 #73 Oct 10 2024 15:29:33
%S A078972 4,6,9,10,14,15,21,25,35,49,121,143,169,187,209,221,247,253,289,299,
%T A078972 319,323,341,361,377,391,403,407,437,451,473,481,493,517,527,529,533,
%U A078972 551,559,583,589,611,629,649,667,671,689,697,703,713,731,737,767,779,781
%N A078972 Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits.
%C A078972 "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]
%C A078972 Up to 10^8 the approximate sum of reciprocals is ~1.232884485... - _Jason Earls_, Oct 15 2010
%C A078972 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
%D A078972 P. D. James, The Private Patient, Knopf, NY, 2008, p. 192. (from _N. J. A. Sloane_, Aug 27 2009)
%H A078972 T. D. Noe, <a href="/A078972/b078972.txt">Table of n, a(n) for n = 1..10537</a>
%H A078972 Dario Alpern, <a href="https://www.alpertron.com.ar/BRILLIANT.HTM">Brilliant Numbers</a>.
%H A078972 Christian N. K. Anderson, <a href="/A078972/a078972.gif">Ulam Spiral</a> of n=1..3000.
%F A078972 a(n) = A239585(n) * A239586(n). - _Reinhard Zumkeller_, Mar 22 2014
%e A078972 1711 = 29*59 is in the sequence since both of its factors have two digits.
%t A078972 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 *)
%t A078972 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 *)
%t A078972 Select[Range@1000, Differences@IntegerLength@Flatten@(ConstantArray@@#&/@FactorInteger[#]) == {0} &] (* _Hans Rudolf Widmer_, Oct 25 2022 *)
%t A078972 dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors *)
%t A078972 Flatten[Array[dlist2, 2]] (* _Paolo Xausa_, Oct 05 2024 *)
%o A078972 (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
%o A078972 (Haskell)
%o A078972 import Data.Function (on)
%o A078972 a078972 n = a078972_list !! (n-1)
%o A078972 a078972_list = filter brilliant a001358_list where
%o A078972    brilliant x = (on (==) a055642) p (x `div` p) where p = a020639 x
%o A078972 -- _Reinhard Zumkeller_, Nov 10 2013, Mar 22 2014
%o A078972 (Python)
%o A078972 from sympy import sieve
%o A078972 A078972 = []
%o A078972 for n in range(3):
%o A078972     pr = list(sieve.primerange(10**n,10**(n+1)))
%o A078972     for i,p in enumerate(pr):
%o A078972         for q in pr[i:]:
%o A078972             A078972.append(p*q)
%o A078972 A078972 = sorted(A078972)
%o A078972 # _Chai Wah Wu_, Aug 26 2014
%Y A078972 Cf. A001358, A085647, A084126, A084127, A055642, A085721.
%K A078972 base,easy,nonn
%O A078972 1,1
%A A078972 _Jason Earls_, Jan 12 2003
%E A078972 Edited by _N. J. A. Sloane_, Aug 27 2009

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