

A145569


Multiples of 6 appear in pairs.


1



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

0,3


COMMENTS

Partial sums of A109720 (characteristic sequence for relative primality to 7).
The first member of the pair of k*6 appears for n = 7*k1; 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.
First differences are periodic: [1,1,1,1,1,1,0].  Paolo P. Lava, Feb 11 2009


REFERENCES

B. Fine and G. Rosenberger, Number theory: an introduction via the distribution of primes, Birkhäuser, Boston, Basel, Berlin, 2007.


LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1000


FORMULA

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_{d7} 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)/((1x^7)*(1x)).
a(n) = (1/49)*Sum_{k=0..n}(9*(k mod 7) + 2*((k+1) mod 7) + 2*((k+2) mod 7) + 2*((k+3) mod 7) + 2*((k+4) mod 7) + 2*((k+5) mod 7)  5*((k+6) mod 7)), with n >= 0.  Paolo P. Lava, Feb 11 2009
a(n) = ceiling(6*n/7) = n  floor(n/7).  Wesley Ivan Hurt, Sep 29 2017


MATHEMATICA

Table[Sum[MoebiusMu[d] Floor[n/d], {d, Divisors[7]}], {n, 0, 74}] (* Indranil Ghosh, Mar 15 2017 *)


PROG

(PARI) for(n=0, 74, print1(sumdiv(7, d, moebius(d) * floor(n/d)), ", ")) \\ Indranil Ghosh, Mar 15 2017


CROSSREFS

Cf. A008683, A109720.
Sequence in context: A006164 A053758 A303789 * A213851 A172475 A171970
Adjacent sequences: A145566 A145567 A145568 * A145570 A145571 A145572


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang Feb 05 2009


STATUS

approved



