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A240670 Numbers n for which all exponents in the prime power factorization of (2*n)! are odious (A000069).

%I A240670 #39 Sep 20 2018 03:04:38
%S A240670 1,3,4,5,15,16
%N A240670 Numbers n for which all exponents in the prime power factorization of (2*n)! are odious (A000069).
%C A240670 The next term, if it exists, must be more than 45000. - _Peter J. C. Moses_, Apr 11 2014
%C A240670 The sequence is finite.
%C A240670 Proof. For sufficiently large n, we always have a prime in (n/4, n/3]. Such primes p divide n! and, at the same time, for them we have 3<=n/p<4. Thus floor(n/p)=3, and in case sqrt(n)<n/4, floor(n/p^2)=0. Therefore, they involve in n! with exponent 3. Since 3 is evil, we are done. Moreover, numerically, using estimate for (4/3)-Ramanujan number (see Shevelev, Greathouse IV, and Moses link, Proposition 8), it is sufficient to consider n>=93 in order for the above arguments to be true. So 16 is the last term of the sequence. - _Vladimir Shevelev_, Apr 11 2014
%H A240670 D. Berend, G. Kolesnik, <a href="http://dx.doi.org/10.1016/j.jnt.2006.08.010">Regularity of patterns in the factorization of n!</a>, J. Number Theory, 124 (2007), no. 1, 181-192.
%H A240670 Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%e A240670 32! = 2^31*3^14*5^7*7^4*11^2*13^2*17*19*23*29*31, and all exponents: 31,14,7,4,2,2,1,1,1,1,1 are odious, so 16 is in the sequence.
%t A240670 odiousQ[n_] := OddQ[DigitCount[n, 2][[1]]];
%t A240670 For[n = 1, True, n++, If[AllTrue[FactorInteger[(2 n)!][[All, 2]], odiousQ], Print[n]]] (* _Jean-François Alcover_, Sep 20 2018 *)
%o A240670 (PARI) isok(n) = {f = factor((2*n)!); sum(i=1, #f~, hammingweight(f[i, 2]) % 2) == #f;} \\ _Michel Marcus_, Apr 11 2014
%Y A240670 Cf. A000069, A240537, A240606, A240619, A240620, A240668, A240669.
%K A240670 nonn,fini,full
%O A240670 1,2
%A A240670 _Vladimir Shevelev_, Apr 10 2014