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

1,2

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

Table of n, a(n) for n=1..54.

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

Cf. A062968, A007305, A002088.

Sequence in context: A309616 A365321 A267452 * A007981 A258847 A238627

Adjacent sequences: A140649 A140650 A140651 * A140653 A140654 A140655

Keyword

easy,nonn

Author

R. J. Mathar, Jul 09 2008

Status

approved