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

3,1

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).

Links

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

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

Example

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.

Maple

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000010, A065464, A127473.

Sequence in context: A088560 A222821 A363418 * A086328 A095997 A274139

Adjacent sequences: A245494 A245495 A245496 * A245498 A245499 A245500

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jul 24 2014

Status

approved