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.

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

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

Table of n, a(n) for n=1..33.

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