A099958 -id:A099958 - cp's OEIS frontend

A049690 a(n) = Sum_{k=1..n} phi(2*k), where phi = Euler totient function, cf. A000010.

0, 1, 3, 5, 9, 13, 17, 23, 31, 37, 45, 55, 63, 75, 87, 95, 111, 127, 139, 157, 173, 185, 205, 227, 243, 263, 287, 305, 329, 357, 373, 403, 435, 455, 487, 511, 535, 571, 607, 631, 663, 703, 727, 769, 809, 833, 877, 923, 955, 997, 1037, 1069, 1117, 1169, 1205

Offset: 0

Clark Kimberling

nonn

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

a(n)=b(2n), where b=A049689. Bisections: A099958, A190815.

Cf. A062570.

A049690 := proc(n) return add(numtheory[phi](2*k), k=1..n): end: seq(A049690(n),n=0..54); # Nathaniel Johnston, May 24 2011

A049690[0]:=0; A049690[n_]:=A049690[n-1]+EulerPhi[2n]; Array[A049690,200,0] (* Enrique Pérez Herrero, Feb 25 2012 *)

a(n)=sum(k=1,n,eulerphi(2*k)) \\ Charles R Greathouse IV, Feb 19 2013

from sympy import totient
def A049690(n): return sum(totient(n) for n in range(1,n+1,2)) + (sum(totient(n) for n in range(2,n+1,2))<<1) # Chai Wah Wu, Aug 04 2024

# faster program using program from A002088 and recursive formula
def A049690(n): return A002088(n) + A049690(n>>1) if n else 0 # Chai Wah Wu, Aug 04 2024

a(n) ~ 4*n^2/Pi^2. - Vaclav Kotesovec, Aug 20 2021

a(n) = A002088(n) + a(floor(n/2)). - Chai Wah Wu, Aug 04 2024

More terms from Vladeta Jovovic, May 18 2001

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

Original entry on oeis.org

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

Hugo Pfoertner, Nov 13 2004

nonn

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.

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.

Bisection of A274401.
Partial sums of A037225.
Cf. A000010, A002088, A099958, A049687, A049690.

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

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

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

A190815 A bisection of A049690.

Original entry on oeis.org

0, 3, 9, 17, 31, 45, 63, 87, 111, 139, 173, 205, 243, 287, 329, 373, 435, 487, 535, 607, 663, 727, 809, 877, 955, 1037, 1117, 1205, 1293, 1385, 1475, 1595, 1695, 1783, 1913, 2005, 2123, 2267, 2379, 2487, 2629, 2763, 2893, 3041, 3177, 3313, 3473, 3625, 3761

Offset: 1

N. J. A. Sloane, May 20 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000010, A049690, A099958.

[n eq 1 select 0 else (&+[EulerPhi(2*j): j in [1..2*(n-1)]]) : n in [1..60] ]; // G. C. Greubel, Dec 03 2023

A190815 := proc(n) option remember: if(n=1)then return 0:fi: return procname(n-1)+numtheory[phi](4*n-6)+numtheory[phi](4*n-4): end: seq(A190815(n),n=1..49); # Nathaniel Johnston, May 24 2011

a[n_] := Sum[EulerPhi[2k], {k, 1, 2n-2}];
Table[a[n], {n, 1, 49}] (* Jean-François Alcover, Apr 16 2023 *)

[sum(euler_phi(2*j) for j in range(1,2*n-1)) for n in range(1,61)] # G. C. Greubel, Dec 03 2023

More terms from Nathaniel Johnston, May 24 2011

cp's OEIS Frontend

A049690 a(n) = Sum_{k=1..n} phi(2*k), where phi = Euler totient function, cf. A000010.

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

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A190815 A bisection of A049690.

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