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A140652 Partial sums of A062968.

%I A140652 #16 Feb 16 2025 08:33:08
%S A140652 1,2,4,6,10,13,19,24,31,38,48,55,67,78,90,102,118,131,149,164,182,201,
%T A140652 223,240,263,286,310,333,361,384,414,441,471,502,534,562,598,633,669,
%U A140652 702,742,777,819,858,898,941,987,1026,1073,1118,1166,1213,1265,1312
%N A140652 Partial sums of A062968.
%C A140652 A062968(n) counts fractions of the format i/j with 1<=j<n and (i,j) relatively prime.
%C A140652 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.
%H A140652 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%F A140652 a(n) = Sum_{i=1..n} A062968(i).
%F A140652 a(n) = Sum_{i=1..n} i - floor(n/(i+1)). - _Wesley Ivan Hurt_, Sep 13 2017
%F A140652 G.f.: x*(2 - x)/(1 - x)^3 - (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^k). - _Ilya Gutkovskiy_, Sep 18 2017
%e A140652 A062968(1)=1 counts the fraction 0/1.
%e A140652 A062968(2)=1 counts 1/2.
%e A140652 A062968(3)=2 counts {1/3,2/3}.
%e A140652 A062968(4)=2 counts {1/4,3/4} skipping 2/4 which could be reduced to 1/2.
%e A140652 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.
%t A140652 Table[n + 1 - DivisorSigma[0, n], {n, 1, 54}] // Accumulate (* _Jean-François Alcover_, Jun 24 2013 *)
%o A140652 (PARI) A062968(n)={ return(n+1-numdiv(n)) ; }
%o A140652 A(n)={ return(sum(i=1,n,A062968(i))) ; }
%o A140652 { for(n=1,100,print1(A(n),", ")) ; }
%Y A140652 Cf. A062968, A007305, A002088.
%K A140652 easy,nonn
%O A140652 1,2
%A A140652 _R. J. Mathar_, Jul 09 2008