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 A133954 Difference between the numbers of nonnegative evil and odious multiples of p_n less than 2^p_n, where p_n = n-th prime. 3
 0, 3, 5, -7, 11, 13, 697, 19, -23, 29, -237367, 37, 97129, 44250483, -47, 53, 59, 61, 67, -71, 1325443061345, -79, 83, 6096136101052865, 6711137545, 101, -103, 107, 197096207419453, 1733616652657, -16388345406766785202757351, 131, 904581545 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). 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. Subset of A225855. LINKS J. Coquet, A summation formula related to the binary digits, Invent. Math. 73 (1983) 107-115. M. Drmota and M. Skalba, Rarified sums of the Thue-Morse sequence, Trans. of the AMS 352 No. 2 (1999) 609-642. D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721. V. Shevelev, On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2, arXiv:0710.1354 [math.NT], 2007. V. Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12. V. Shevelev, Exact exponent of remainder term of Gelfond's digit theorem in the binary case, Acta Arithmetica 136 (2009) 91-100. I. Shparlinski, On the size of the Gelfond exponent, J. of Number Theory, 130, no.4 (2010), 1056-1060. FORMULA 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 EXAMPLE Consider p_2=3; since 0,3,6 are evil, then a(2) = 3 - 0 = 3. PROG (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 CROSSREFS Cf. A000069, A001969, A001122, A139035, A225855. Sequence in context: A108817 A288892 A123677 * A087325 A072151 A332028 Adjacent sequences:  A133951 A133952 A133953 * A133955 A133956 A133957 KEYWORD sign AUTHOR Vladimir Shevelev, Sep 30 2007, Dec 17 2008 EXTENSIONS Extended by Robert Gerbicz, Jan 03 2011 STATUS approved

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Last modified May 13 04:09 EDT 2021. Contains 343836 sequences. (Running on oeis4.)