A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

0, 1, 4, 6, 18, 11, 45, 36, 66, 50, 150, 68, 234, 132, 192, 216, 488, 198, 648, 312, 510, 460, 1089, 420, 1140, 732, 1161, 822, 2044, 616, 2430, 1376, 1810, 1528, 2400, 1260, 3942, 2052, 2880, 2008, 5260, 1644, 5943, 2950, 3672, 3509, 7567, 2736, 7497, 3670

Offset: 1

Author

Zhi-Wei Sun, Oct 14 2015

Keywords

• nonn

Comments

Conjecture: (i) All the terms of this sequence are pairwise distinct.

(ii) All the numbers phi(n)*pi(n*(n-1)) (n = 1,2,3,...) are pairwise distinct.

(iii) All the numbers phi(n^2)*pi(n^2) = n*phi(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct.

We have checked this conjecture via Mathematica. For example, we have verified that a(n) (n = 1..4*10^5) are indeed pairwise distinct.

See also A263325 for a similar conjecture.

Examples

			a(1) = 0 since pi(1^2)*phi(1)/2 = 0*1/2 = 0.
a(2) = 1 since pi(2^2)*phi(2)/2 = 2*1/2 = 1.
a(3) = 4 since pi(3^2)*phi(3)/2 = 4*2/2 = 4.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000290, A000720, A002618, A038107, A263317, A263321, A263325.

Programs

Magma

[#PrimesUpTo(n^2)*EulerPhi(n)/2: n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

Mathematica

a[n_]:=a[n]=PrimePi[n^2]*EulerPhi[n]/2
Do[Print[n," ",a[n]],{n,1,50}]

PARI

a(n) = primepi(n^2)*eulerphi(n)/2; \\ Michel Marcus, Oct 15 2015