A099957 a(n) = Sum_{k=0..n-1} phi(2k+1).

1, 3, 7, 13, 19, 29, 41, 49, 65, 83, 95, 117, 137, 155, 183, 213, 233, 257, 293, 317, 357, 399, 423, 469, 511, 543, 595, 635, 671, 729, 789, 825, 873, 939, 983, 1053, 1125, 1165, 1225, 1303, 1357, 1439, 1503, 1559, 1647, 1719, 1779, 1851, 1947

Offset: 1

Author

Hugo Pfoertner, Nov 13 2004

Keywords

• nonn

Comments

The n-th term is the number of notes of the (2n-1)-limit tonality diamond. This is a term from music theory and means the scale consisting of the rational numbers r, 1 <= r < 2, such that the odd part of both the numerator and the denominator of r, when reduced to lowest terms, is less than or equal to the fixed odd number 2n-1. - Gene Ward Smith, Mar 27 2006

(1/4)*Number of distinct angular positions under which an observer positioned at the center of a square of a square lattice can see the (2n) X (2n) points symmetrically surrounding his position.

(1/8)*number of distinct angular positions under which an observer positioned at a lattice point of a square lattice can see the (2n+1)X(2n+1) points symmetrically surrounding his position gives A002088.

(1/2)*number of distinct angular positions under which an observer positioned at the center of an edge of a square lattice can see the (2n)X(2n-1) points symmetrically surrounding his position gives A099958.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Lv Chuan, On the Mean Value of an Arithmetical Function, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 89-92.
• Wikipedia, Tonality diamond.

Crossrefs

Bisection of A274401.

Partial sums of A037225.

Cf. A000010, A002088, A099958, A049687, A049690.

Programs

Mathematica

Accumulate[EulerPhi[2*Range[0,50]+1]] (* Harvey P. Dale, Aug 20 2021 *)

PARI

apply( {A099957(n)=sum(k=1,n, eulerphi(2*k-1))}, [1..55]) \\ M. F. Hasler, Apr 03 2023

Formula

a(n+1) - a(n) = phi(2n+1) (A037225).

a(n) = (8/Pi^2)*n^2 + O(n^(3/2+eps)) (Lemma 1 in Lv Chuan, 2004). - Amiram Eldar, Aug 02 2022, corrected by M. F. Hasler, Mar 26 2023

a(n) = A002088(2*n-1) - A049690(n-1). - Chai Wah Wu, Aug 04 2024