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A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x. 3
0, 6, 16, 42, 54, 132, 120, 270, 286, 450, 360, 952, 546, 1056, 1152, 1674, 1098, 2574, 1440, 3276, 2720, 3312, 2376, 6300, 3534, 5124, 5160, 7672, 4380, 11088, 5184, 10836, 8688, 10314, 9600, 19110, 8322, 13680, 13440, 22590, 11046, 26304, 12452, 24780, 23868, 22968, 15792, 42408, 20349, 34131 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: (i) All the terms of this sequence are pairwise distinct.

(ii) All the numbers sigma(n)*pi(n*(n+1)) (n = 1,2,3,...) are pairwise distinct.

(iii) All the numbers n*sigma(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct, and all the numbers sigma(n^2)*pi(n^2) (n = 1,2,3,...) are also pairwise distinct.

(iv) All the numbers n*phi(n)*sigma(n^2) = phi(n^2)*sigma(n^2) (n = 1,2,3,...) are pairwise distinct, where phi(.) is Euler's totient function.

We have verified that the terms a(n) (n = 1..4*10^5) are indeed pairwise distinct.

See also A263319 for a similar conjecture.

LINKS

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

EXAMPLE

a(1) = 0 since sigma(1)*pi(1^2) = 1*0 = 0.

a(2) = 6 since sigma(2)*pi(2^2) = 3*2 = 6.

MATHEMATICA

a[n_]:=a[n]=DivisorSigma[1, n]*PrimePi[n^2]

Do[Print[n, " ", a[n]], {n, 1, 50}]

PROG

(PARI) a(n) = sigma(n)*primepi(n^2); \\ Michel Marcus, Oct 15 2015

(MAGMA) [#PrimesUpTo(n^2)*SumOfDivisors(n): n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

CROSSREFS

Cf. A000010, A000203, A000290, A000720, A002618, A038107, A065764, A263317, A263319.

Sequence in context: A073570 A283960 A283330 * A107614 A317758 A010915

Adjacent sequences:  A263322 A263323 A263324 * A263326 A263327 A263328

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 14 2015

STATUS

approved

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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)