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%I #25 Sep 08 2022 08:46:14
%S 0,1,4,6,18,11,45,36,66,50,150,68,234,132,192,216,488,198,648,312,510,
%T 460,1089,420,1140,732,1161,822,2044,616,2430,1376,1810,1528,2400,
%U 1260,3942,2052,2880,2008,5260,1644,5943,2950,3672,3509,7567,2736,7497,3670
%N 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.
%C Conjecture: (i) All the terms of this sequence are pairwise distinct.
%C (ii) All the numbers phi(n)*pi(n*(n-1)) (n = 1,2,3,...) are pairwise distinct.
%C (iii) All the numbers phi(n^2)*pi(n^2) = n*phi(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct.
%C We have checked this conjecture via Mathematica. For example, we have verified that a(n) (n = 1..4*10^5) are indeed pairwise distinct.
%C See also A263325 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A263319/b263319.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 0 since pi(1^2)*phi(1)/2 = 0*1/2 = 0.
%e a(2) = 1 since pi(2^2)*phi(2)/2 = 2*1/2 = 1.
%e a(3) = 4 since pi(3^2)*phi(3)/2 = 4*2/2 = 4.
%t a[n_]:=a[n]=PrimePi[n^2]*EulerPhi[n]/2
%t Do[Print[n," ",a[n]],{n,1,50}]
%o (PARI) a(n) = primepi(n^2)*eulerphi(n)/2; \\ _Michel Marcus_, Oct 15 2015
%o (Magma) [#PrimesUpTo(n^2)*EulerPhi(n)/2: n in [1..80]]; // _Vincenzo Librandi_, Oct 15 2015
%Y Cf. A000010, A000290, A000720, A002618, A038107, A263317, A263321, A263325.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Oct 14 2015
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