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URL: https://oeis.org/A263319

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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.
4
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.
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 14 2015
STATUS
approved