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

Author

Zhi-Wei Sun, Oct 14 2015

Keywords

• nonn

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.

Examples

			a(1) = 0 since sigma(1)*pi(1^2) = 1*0 = 0.
a(2) = 6 since sigma(2)*pi(2^2) = 3*2 = 6.
		

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

a[n_]:=a[n]=DivisorSigma[1,n]*PrimePi[n^2]
Do[Print[n," ",a[n]],{n,1,50}]

PARI

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