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A099957 a(n) = Sum_{k=0..n-1} phi(2k+1). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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.
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
MATHEMATICA
Accumulate[EulerPhi[2*Range[0, 50]+1]] (* Harvey P. Dale, Aug 20 2021 *)
PROG
(PARI) apply( {A099957(n)=sum(k=1, n, eulerphi(2*k-1))}, [1..55]) \\ M. F. Hasler, Apr 03 2023
CROSSREFS
Bisection of A274401.
Partial sums of A037225.
Sequence in context: A147614 A171747 A031215 * A086148 A262086 A205956
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Nov 13 2004
STATUS
approved

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Last modified March 29 08:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)