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

1,3

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.

Links

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

Example

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.

Mathematica

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

Prog

(PARI) a(n) = primepi(n^2)*eulerphi(n)/2; \\ Michel Marcus, Oct 15 2015
(Magma) [#PrimesUpTo(n^2)*EulerPhi(n)/2: n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

Crossrefs

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

Sequence in context: A332986 A113610 A309334 * A062046 A102020 A125133

Adjacent sequences: A263316 A263317 A263318 * A263320 A263321 A263322

Keyword

nonn

Author

Zhi-Wei Sun, Oct 14 2015

Status

approved