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Number of prime divisors of n-th term of Sylvester's sequence A000058.
7

%I #24 Oct 21 2023 16:38:13

%S 1,1,1,1,2,1,4,4,3,5,4

%N Number of prime divisors of n-th term of Sylvester's sequence A000058.

%C All numbers less than 2.5*10^15 in Sylvester's sequence are squarefree and no squareful numbers in this sequence are known (Vardi 1991).

%C a(n) is currently unknown for all n > 10. - _Jens Kruse Andersen_, Jun 19 2014

%D Ilan Vardi, "Are All Euclid Numbers Squarefree?" and "PowerMod to the Rescue", Sections 5.1 and 5.2 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 82-89, 1991.

%H Jens Kruse Andersen, <a href="http://primerecords.dk/sylvester-factors.htm">Factorization of Sylvester's sequence</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvestersSequence.html">Sylvester's sequence</a>

%F a(n) = A001221(A000058(n)).

%e a(8) = 3 because A000058(8) = 5295435634831 * 31401519357481261 * 77366930214021991992277 is a product of 3 primes.

%e a(9) = 5 because A000058(9) = 181 * 1987 * 112374829138729 * 114152531605972711 * 35874380272246624152764569191134894955972560447869169859142453622851 is the product of 5 prime factors

%e a(10) = 4 because A000058(10) = 2287 * 2271427 * 21430986826194127130578627950810640891005487 * P156 is the product of 4 prime factors.

%e Here P156 = 24605022397522123277426691306421099608611770732459695261246331125\

%e 73460100430857224101455594897691626456909430029315374035313628946949460093682\

%e 49974883220589.

%t PrimeNu[NestList[#^2 - # + 1 &, 2, 7]] (* _G. C. Greubel_, May 09 2017 *)

%Y Cf. A000058, A001221, A014546, A091336.

%K hard,more,nonn

%O 0,5

%A _Max Alekseyev_, Dec 30 2003

%E a(9) from _T. D. Noe_, Dec 31 2003

%E a(10) from _Ken Takusagawa_, Apr 11 2006