A135177 - OEIS

%I #54 Dec 14 2024 17:46:22

%S 4,18,100,294,1210,2028,4624,6498,11638,23548,28830,49284,67240,77658,

%T 101614,146068,201898,223260,296274,352870,383688,486798,564898,

%U 697048,903264,1020100,1082118,1213594,1283148,1430128,2032254,2230930

%N a(n) = p^2*(p-1), where p = prime(n).

%H Antti Karttunen, <a href="/A135177/b135177.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Vincenzo Librandi)

%H <a href="/index/Pri#prime_powers">Index to sequences related to prime powers</a>.

%F a(n) = p^3 - p^2 = A030078(n) - A001248(n).

%F a(n) = A000010(prime(n)^3). - _R. J. Mathar_, Oct 15 2017

%F Sum_{n>=1} 1/a(n) = A152441. - _Amiram Eldar_, Nov 09 2020

%F From _Amiram Eldar_, Nov 22 2022: (Start)

%F Product_{n>=1} (1 + 1/a(n)) = A065483.

%F Product_{n>=1} (1 - 1/a(n)) = A065414. (End)

%F a(n) = 2*A138416(n). - _Antti Karttunen_, Dec 14 2024

%e a(4) = 294 because the 4th prime number is 7, 7^2 = 49, 7-1 = 6 and 49 * 6 = 294.

%t Table[Prime[n]^3-Prime[n]^2, {n, 1, 12}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 29 2008 *)

%t Table[p^3-p^2,{p,Prime[Range[40]]}] (* _Harvey P. Dale_, Jan 15 2015 *)

%o (Magma)[(p^3-p^2): p in PrimesUpTo(200)]; // _Vincenzo Librandi_, Dec 15 2010

%o (PARI) forprime(p=2,1e3,print1(p^2*(p-1)", ")) \\ _Charles R Greathouse IV_, Jun 16 2011

%o (PARI) A135177(n) = eulerphi(prime(n)^3); \\ _Antti Karttunen_, Dec 14 2024

%o (PARI) A135177(n) = ((p->(p-1)*p*p)(prime(n))); \\ _Antti Karttunen_, Dec 14 2024

%Y Cf. A001248 (p^2), A030078 (p^3), A045991 (n^2 * (n-1)), A065414, A065483, A138416 (terms halved), A152441.

%Y Column 4 of A379010.

%K nonn,easy

%O 1,1

%A _Omar E. Pol_, Nov 25 2007