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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified February 26 05:55 EST 2024. Contains 370335 sequences. (Running on oeis4.)