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A140652
Partial sums of A062968.
0
1, 2, 4, 6, 10, 13, 19, 24, 31, 38, 48, 55, 67, 78, 90, 102, 118, 131, 149, 164, 182, 201, 223, 240, 263, 286, 310, 333, 361, 384, 414, 441, 471, 502, 534, 562, 598, 633, 669, 702, 742, 777, 819, 858, 898, 941, 987, 1026, 1073, 1118, 1166, 1213, 1265, 1312
OFFSET
1,2
COMMENTS
A062968(n) counts fractions of the format i/j with 1<=j<n and (i,j) relatively prime.
The partial sum gives the number of "essentially" distinct values on the unit circle for all roots up to the n-th. This relates to the problem of decomposing the generating function of the restricted partitions of n, A026820, into partial fractions.
LINKS
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.
FORMULA
a(n) = Sum_{i=1..n} A062968(i).
a(n) = Sum_{i=1..n} i - floor(n/(i+1)). - Wesley Ivan Hurt, Sep 13 2017
G.f.: x*(2 - x)/(1 - x)^3 - (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017
EXAMPLE
A062968(1)=1 counts the fraction 0/1.
A062968(2)=1 counts 1/2.
A062968(3)=2 counts {1/3,2/3}.
A062968(4)=2 counts {1/4,3/4} skipping 2/4 which could be reduced to 1/2.
A062968(5)=4 counts {1/5,2/5,3/5,4/5}. The value a(5)=1+1+2+2+4=10 counts all these distinct fractions {0/1,1/2,1/3,2/3,..,4/5}, which represent the phases of the roots of the polynomials 1-x^j, j=1..5.
MATHEMATICA
Table[n + 1 - DivisorSigma[0, n], {n, 1, 54}] // Accumulate (* Jean-François Alcover, Jun 24 2013 *)
PROG
(PARI) A062968(n)={ return(n+1-numdiv(n)) ; }
A(n)={ return(sum(i=1, n, A062968(i))) ; }
{ for(n=1, 100, print1(A(n), ", ")) ; }
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
R. J. Mathar, Jul 09 2008
STATUS
approved