A099957 - OEIS

%I #36 Aug 06 2024 07:24:08

%S 1,3,7,13,19,29,41,49,65,83,95,117,137,155,183,213,233,257,293,317,

%T 357,399,423,469,511,543,595,635,671,729,789,825,873,939,983,1053,

%U 1125,1165,1225,1303,1357,1439,1503,1559,1647,1719,1779,1851,1947

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

%C 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

%C (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.

%C (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.

%C (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.

%H Seiichi Manyama, <a href="/A099957/b099957.txt">Table of n, a(n) for n = 1..10000</a>

%H Lv Chuan, <a href="https://citeseerx.ist.psu.edu/pdf/134f67dafd17bab61928c5a02e2e9808a27a1dad">On the Mean Value of an Arithmetical Function</a>, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 89-92.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tonality_diamond">Tonality diamond</a>.

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

%F 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

%F a(n) = A002088(2*n-1) - A049690(n-1). - _Chai Wah Wu_, Aug 04 2024

%t Accumulate[EulerPhi[2*Range[0,50]+1]] (* _Harvey P. Dale_, Aug 20 2021 *)

%o (PARI) apply( {A099957(n)=sum(k=1,n, eulerphi(2*k-1))}, [1..55]) \\ _M. F. Hasler_, Apr 03 2023

%Y Bisection of A274401.

%Y Partial sums of A037225.

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

%K nonn

%O 1,2

%A _Hugo Pfoertner_, Nov 13 2004