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%I #45 Sep 08 2022 08:46:06
%S 1,2,2,3,4,3,5,4,2,5,5,4,3,1,2,3,6,4,3,6,4,2,4,5,2,4,3,6,7,7,4,3,2,4,
%T 5,3,4,7,4,6,6,4,1,4,5,4,6,6,5,3,6,4,6,6,4,11,6,6,6,4,5,5,2,6,7
%N Number of prime divisors of (10^(n+3) + 666)*10^(n+1) + 1 (see A232449) counted with multiplicity.
%C The Belphegor numbers (A232449), though large and rarely prime (A232448), tend to contain only very few prime factors. One wonders whether this sequence might be bounded.
%C From _Robert Israel_, Feb 23 2017: (Start)
%C The sequence is unbounded.
%C Indeed, if p is in A001913 such that the polynomial 10^4 x^2 + 6660 x + 1 has a simple root mod p, then for all k there exist Belphegor numbers divisible by p^k.
%C For example, p=29 works; we have A232449(n) divisible by 29^k for n = 6, 158, 5522, 41570, 8153130, 107172470, 3553045502, 136793469406, 2761185750502, 142830181379582, ...
%C (End)
%H FactorDB, <a href="http://factordb.com/?query=%2810%5E%28n%2B3%29%2B666%29*10%5E%28n%2B1%29%2B1">(10^(n+3)+666)*10^(n+1)+1</a>.
%H Clifford A. Pickover, <a href="http://sprott.physics.wisc.edu/pickover/pc/1000000000000066600000000000001.html">Belphegor's Prime: 1000000000000066600000000000001</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Belphegor%27s_prime">Belphegor's prime</a>
%p seq(numtheory:-bigomega(10^(2*n+4)+666*10^(n+1)+1), n=0..30); # _Robert Israel_, Feb 23 2017
%t Table[Total[Transpose[FactorInteger[(10^(n + 3) + 666)*10^(n + 1) + 1]][[2]]], {n, 0, 25}] (* _T. D. Noe_, Nov 28 2013 *)
%o (PARI) a(n)=bigomega(10^(n+1)*(10^(n+3)+666)+1) \\ _Charles R Greathouse IV_, Nov 26 2013
%o (Magma) [&+[p[2]: p in Factorization(666*10^(n+1)+100^(n+2)+1)]: n in [0..40]]; // _Bruno Berselli_, Nov 27 2013
%Y Cf. A001913, A232448 (indices of Belphegor primes), A232449 (Belphegor numbers), A232450 (largest prime factor of A232449(n)).
%K nonn,more
%O 0,2
%A _Stanislav Sykora_, Nov 24 2013
%E a(45)-a(64) from _Amiram Eldar_, Apr 11 2020
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