A245497 a(n) = phi(n)^2/2, where phi(n) = A000010(n), the Euler totient function.

2, 2, 8, 2, 18, 8, 18, 8, 50, 8, 72, 18, 32, 32, 128, 18, 162, 32, 72, 50, 242, 32, 200, 72, 162, 72, 392, 32, 450, 128, 200, 128, 288, 72, 648, 162, 288, 128, 800, 72, 882, 200, 288, 242, 1058, 128, 882, 200, 512, 288, 1352, 162, 800, 288, 648, 392, 1682

Offset: 3

Author

Wesley Ivan Hurt, Jul 24 2014

Keywords

• nonn
• easy

Comments

Values of a(n) < 3 are non-integers since phi(1) = phi(2) = 1 (odd). Since phi(n) is even for all n > 2, a(n) is a positive integer.

a(n) gives the sum of all the parts in the partitions of phi(n) with exactly two parts (see example).

a(n) is also the area of a square with diagonal phi(n).

Examples

			a(5) = 8; since phi(5)^2/2 = 4^2/2 = 8. The partitions of phi(5) = 4 into exactly two parts are: (3,1) and (2,2). The sum of all the parts in these partitions gives: 3+1+2+2 = 8.
a(7) = 18; since phi(7)^2/2 = 6^2/2 = 18. The partitions of phi(7) = 6 into exactly two parts are: (5,1), (4,2) and (3,3). The sum of all the parts in these partitions gives: 5+1+4+2+3+3 = 18.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A000010, A065464, A127473.

Programs

Maple

with(numtheory): 245497:=n->phi(n)^2/2: seq(245497(n), n=3..50);

Mathematica

Table[EulerPhi[n]^2/2, {n, 3, 50}]

PARI

vector(100, n, eulerphi(n+2)^2/2) \\ Derek Orr, Aug 04 2014

Formula

a(n) = phi(n)^2/2 = A000010(n)^2/2 = A127473(n)/2, n > 2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/6) * Product_{p prime} (1 - (2*p-1)/p^3) = A065464 / 6 = 0.0713749... . - Amiram Eldar, Nov 14 2024