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%I #27 Mar 17 2024 02:10:47
%S 0,1,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,18,19,20,21,22,
%T 23,24,24,25,26,27,28,29,30,30,31,32,33,34,35,36,36,37,38,39,40,41,42,
%U 42,43,44,45,46,47,48,48,49,50,51,52,53,54,54,55,56,57,58,59,60,60,61,62,63,64
%N Multiples of 6 appear in pairs.
%C Partial sums of A109720 (characteristic sequence for relative primality to 7).
%C The first member of the pair of k*6 appears for n = 7*k-1; put a(-1)=0.
%C This is the member m=7 in the family of sequences, called N(m), which are partial sums of Xi(m). Xi(7) = A109720.
%D B. Fine and G. Rosenberger, Number theory: an introduction via the distribution of primes, Birkhäuser, Boston, Basel, Berlin, 2007.
%H Indranil Ghosh, <a href="/A145569/b145569.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = number of positive numbers <= n relatively prime to 7, n >= 1. a(0)=1.
%F a(n) = Sum_{k=1..n} A109720(k), n >= 1.
%F a(n) = Sum_{d|7} mu(d)*floor(n/d), n >= 1, with the Moebius function mu(n) = A008683(n) (Legendre formula for the sieve of Eratosthenes (here for m=7). See, e.g., the Fine and Rosenberger reference, p. 200.
%F O.g.f.: x*(1+x+x^2+x^3+x^4+x^5)/((1-x^7)*(1-x)).
%F a(n) = ceiling(6*n/7) = n - floor(n/7). - _Wesley Ivan Hurt_, Sep 29 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (15 - 4*sqrt(3))*Pi/36. - _Amiram Eldar_, Sep 30 2022
%t Table[Sum[MoebiusMu[d] Floor[n/d], {d, Divisors[7]}], {n, 0,74}] (* _Indranil Ghosh_, Mar 15 2017 *)
%o (PARI) for(n=0, 74, print1(sumdiv(7, d, moebius(d) * floor(n/d)),", ")) \\ _Indranil Ghosh_, Mar 15 2017
%Y Cf. A008683, A109720.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_ Feb 05 2009
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