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A145569
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Multiples of 6 appear in pairs.
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1
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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, 23, 24, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 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
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OFFSET
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0,3
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COMMENTS
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Partial sums of A109720 (characteristic sequence for relative primality to 7).
The first member of the pair of k*6 appears for n = 7*k-1; put a(-1)=0.
This is the member m=7 in the family of sequences, called N(m), which are partial sums of Xi(m). Xi(7) = A109720.
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REFERENCES
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B. Fine and G. Rosenberger, Number theory: an introduction via the distribution of primes, Birkhäuser, Boston, Basel, Berlin, 2007.
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LINKS
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FORMULA
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a(n) = number of positive numbers <= n relatively prime to 7, n >= 1. a(0)=1.
a(n) = Sum_{k=1..n} A109720(k), n >= 1.
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.
O.g.f.: x*(1+x+x^2+x^3+x^4+x^5)/((1-x^7)*(1-x)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (15 - 4*sqrt(3))*Pi/36. - Amiram Eldar, Sep 30 2022
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MATHEMATICA
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Table[Sum[MoebiusMu[d] Floor[n/d], {d, Divisors[7]}], {n, 0, 74}] (* Indranil Ghosh, Mar 15 2017 *)
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PROG
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(PARI) for(n=0, 74, print1(sumdiv(7, d, moebius(d) * floor(n/d)), ", ")) \\ Indranil Ghosh, Mar 15 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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