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.

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

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