A140652 Partial sums of A062968.

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

Author

R. J. Mathar, Jul 09 2008

Keywords

• easy
• nonn

Comments

A062968(n) counts fractions of the format i/j with 1<=j

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.

Crossrefs

Cf. A062968, A007305, A002088.

Programs

Mathematica

Table[n + 1 - DivisorSigma[0, n], {n, 1, 54}] // Accumulate (* Jean-François Alcover, Jun 24 2013 *)

PARI

A062968(n)={ return(n+1-numdiv(n)) ; }
A(n)={ return(sum(i=1,n,A062968(i))) ; }
{ for(n=1,100,print1(A(n),", ")) ; }

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

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