A263325 - OEIS

%I #16 Sep 08 2022 08:46:14

%S 0,6,16,42,54,132,120,270,286,450,360,952,546,1056,1152,1674,1098,

%T 2574,1440,3276,2720,3312,2376,6300,3534,5124,5160,7672,4380,11088,

%U 5184,10836,8688,10314,9600,19110,8322,13680,13440,22590,11046,26304,12452,24780,23868,22968,15792,42408,20349,34131

%N 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.

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

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

%C (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.

%C (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.

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

%C See also A263319 for a similar conjecture.

%H Zhi-Wei Sun, <a href="/A263325/b263325.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

%o (PARI) a(n) = sigma(n)*primepi(n^2); \\ _Michel Marcus_, Oct 15 2015

%o (Magma) [#PrimesUpTo(n^2)*SumOfDivisors(n): n in [1..80]]; // _Vincenzo Librandi_, Oct 15 2015

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

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Oct 14 2015