A272173 - OEIS

%I #35 Sep 08 2022 08:46:16

%S 3,12,24,56,72,168,144,300,312,540,384,1064,588,1056,1152,1674,1080,

%T 2418,1360,3024,2368,2880,2016,5400,3038,4284,4160,6048,3300,8208,

%U 4096,8316,6624,7560,7200,13832,6004,9840,9408,15660,7560,17472,8448,16296,15444,14400,10176,27776

%N Product of the sum of the divisors of n and the sum of the divisors of n-th prime.

%C Numbers that occur twice in the sequence include 7560, 816000, 2709504, 31752000. Are there infinitely many? Does any number occur more than twice? - _Robert Israel_, Sep 12 2018

%H Robert Israel, <a href="/A272173/b272173.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = sigma(n)*sigma(prime(n)) = sigma(n)*(1 + prime(n)) = A000203(n)*(1 + A000040(n)) = A000203(n)*A008864(n).

%F a(n) = sigma(n*prime(n)) = A000203(n*A000040(n)) = A000203(A033286(n)).

%F a(n) = A000203(n) + A272211(n).

%e For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13, and the 9th prime is 23, and the sum of the divisors of 23 is 1 + 23 = 24, and 13*24 = 312, so a(9) = 312.

%e On the other hand 9*23 = 207, and the sum of the divisors of 207 is 1 + 3 + 9 + 23 + 69 + 207 = 312, so a(9) = 312.

%p f:= n -> numtheory:-sigma(n)*(1+ithprime(n)):

%p map(f, [$1..100]); # _Robert Israel_, Sep 12 2018

%t Table[DivisorSigma[1, n]*DivisorSigma[1, Prime[n]], {n, 1, 50}] (* _G. C. Greubel_, Apr 27 2016 *)

%o (PARI) a(n) = sigma(n)*sigma(prime(n)); \\ _Michel Marcus_, Apr 27 2016

%o (Magma) [SumOfDivisors(n)*SumOfDivisors(NthPrime(n)): n in [1..50]]; // _Vincenzo Librandi_, Sep 13 2018

%Y Cf. A000040, A000203, A008864, A033286, A272211.

%K nonn

%O 1,1

%A _Omar E. Pol_, Apr 26 2016