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%I #34 Mar 10 2015 02:01:08
%S 0,3,5,-7,11,13,697,19,-23,29,-237367,37,97129,44250483,-47,53,59,61,
%T 67,-71,1325443061345,-79,83,6096136101052865,6711137545,101,-103,107,
%U 197096207419453,1733616652657,-16388345406766785202757351,131,904581545
%N Difference between the numbers of nonnegative evil and odious multiples of p_n less than 2^p_n, where p_n = n-th prime.
%C The following statements are true: 1) If prime p_n has a primitive root 2, then a(n)=p_n; 2) If prime p_n has a semiprimitive root 2, then a(n)=-p_n (for definition of semiprimitive root 2 of a prime, see the 2nd link, p. 1).
%C A comparison of Gerbicz's calculations up to a(46) with A001122 and A139035 shows that one can conjecture that the converse statements are true as well.
%C Subset of A225855.
%H J. Coquet, <a href="http://dx.doi.org/10.1007/BF01393827">A summation formula related to the binary digits</a>, Invent. Math. 73 (1983) 107-115.
%H M. Drmota and M. Skalba, <a href="http://dx.doi.org/10.1090/S0002-9947-99-02277-1">Rarified sums of the Thue-Morse sequence</a>, Trans. of the AMS 352 No. 2 (1999) 609-642.
%H D. J. Newman, <a href="http://dx.doi.org/10.1090/S0002-9939-1969-0244149-8">On the number of binary digits in a multiple of three</a>, Proc. Amer. Math. Soc. 21 (1969) 719-721.
%H V. Shevelev, <a href="http://arxiv.org/abs/0710.1354">On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2</a>, arXiv:0710.1354 [math.NT], 2007.
%H V. Shevelev, <a href="http://www.hindawi.com/journals/ijmms/2008/908045.html">Generalized Newman phenomena and digit conjectures on primes</a>, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.
%H V. Shevelev, <a href="http://dx.doi.org/10.4064/aa136-1-7">Exact exponent of remainder term of Gelfond's digit theorem in the binary case</a>, Acta Arithmetica 136 (2009) 91-100.
%H I. Shparlinski, <a href="http://dx.doi.org/10.1016/j.jnt.2009.09.005">On the size of the Gelfond exponent</a>, J. of Number Theory, 130, no.4 (2010), 1056-1060.
%F a(n) = p_n if 2 is a primitive root of p_n (A001122); a(n) = -p_n if p_n is in A139035, i.e., -2 is a primitive root of p_n [Shevelev, 2007]. No other exact regularity of the sequence is known until now. - _Vladimir Shevelev_, Oct 26 2014
%e Consider p_2=3; since 0,3,6 are evil, then a(2) = 3 - 0 = 3.
%o (PARI) a(p)=o=e=vector(p,i,0);e[p]=1;r=1;for(i=1,p,o2=e2=vector(p);for(j=1,p,w=(j-r)%p;if(w==0,w=p);o2[j]=o[j]+e[w];e2[j]=e[j]+o[w]);o=o2;e=e2;r=(2*r)%p);return(e[p]-o[p]) \\ _Robert Gerbicz_, Jan 03 2011
%Y Cf. A000069, A001969, A001122, A139035, A225855.
%K sign
%O 1,2
%A _Vladimir Shevelev_, Sep 30 2007, Dec 17 2008
%E Extended by _Robert Gerbicz_, Jan 03 2011
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