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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 March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)