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%I #11 Apr 09 2012 17:04:57
%S 19,131,34,19,35,35,35,67,259,575,67,67,67,131,259,515,1027,131,131,
%T 131,131,131,259,259,259,514,515,515,515,8195
%N The maximal positive integer m for which the exponents of 2 and prime(n) in the prime power factorization of m! are both powers of 2.
%C Or a(n) is the maximal m for which the Fermi-Dirac representation of m! (see comment in A050376) contains single power of 2 and single power of prime(n).
%H V. Shevelev, <a href="http://journals.impan.gov.pl/aa/Inf/126-3-1.html">Compact integers and factorials</a>, Acta Arithmetica 126 (2007), no. 3, 195-236.
%F a(2)=19, a(3)=131; if prime(n) has the form (2^(4k+1)+3)/5 for k>=1,then a(n)=5*prime(n)-1; if prime(n)>=17 is Fermat prime, then a(n)=2*prime(n)+1; if prime(n) has the form 2^k+3 for k>=3, then a(n)=2*prime(n)-3; otherwise, if prime(n) is in interval [2^(k-1)+5, 2^k) for k>=4, then a(n)=3+2^(k+floor(log_2((p_n-5)/(2^k-prime(n)))). In any case, a(n)<=(1/2)*(prime(n)+1)^2+3. Equality holds for Mersenne primes>=31.
%e For n=31, prime(n)=127 is Mersenne primes. Thus a(31)=(1/2)*128^2+3=8195.
%Y Cf. A000142, A177436, A177378, A177355, A177349, A177458, A177498, A050376, A169655, A169661.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, May 09 2010
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