Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #27 Sep 08 2022 08:46:12
%S 0,12,64,180,384,700,1152,1764,2560,3564,4800,6292,8064,10140,12544,
%T 15300,18432,21964,25920,30324,35200,40572,46464,52900,59904,67500,
%U 75712,84564,94080,104284,115200,126852,139264,152460,166464,181300,196992,213564
%N a(n) = (4*n+8)*n^2.
%C Let r be a natural number such that r has 17 proper divisors and 5 prime factors (note that these prime factors do not have to be distinct). The difference between these two values, say d(r), is in this case 12. Where n is a positive integer, d(r^n)=(4*n+8)*n^2.
%C The integers that satisfy the proper-divisor-prime-factor requirement are those of A179643.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 4*A152618(n+1).
%F G.f.: 4*x*(3+4*x-x^2)/(1-x)^4. - _Vincenzo Librandi_, Jun 06 2015
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Vincenzo Librandi_, Jun 06 2015
%F For any m, let x=A179643(m), then a(n) = A000005(x^n) - A001222(x^n). - _Michel Marcus_, Jul 09 2015
%e The smallest integer that satisfies the (17, 5) requirement is 180: it has 17 proper divisors (1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90) and 5 prime factors (2, 2, 3, 3, 5), so d(120)=12=a(1).
%e The square of 180, 32400, we would expect to have a difference of 64 between the number of its proper divisors and prime factors, and with respectively 74 and 10, d(32400)=64=a(2) indeed. Checking this with further integer powers of 180 will continue to generate terms in this sequence.
%t Table[(4 n + 8) n^2, {n, 0, 40}] (* or *) CoefficientList[Series[4 x (3 + 4 x - x^2)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2015 *)
%o (Magma) [(4*n+8)*n^2: n in [0..50]] /* or */ I:=[0, 12, 64, 180]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 06 2015
%o (PARI) vector(50,n,n--;(4*n+8)*n^2) \\ _Derek Orr_, Jun 21 2015
%Y Cf. A152618, A179643.
%K nonn,easy
%O 0,2
%A _Garrett Frandson_, Jun 05 2015
%E More terms from _Vincenzo Librandi_, Jun 06 2015