This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A258402 #45 Nov 14 2024 00:06:55
%S A258402 0,11,72,243,608,1275,2376,4067,6528,9963,14600,20691,28512,38363,
%T A258402 50568,65475,83456,104907,130248,159923,194400,234171,279752,331683,
%U A258402 390528,456875,531336,614547,707168,809883,923400,1048451,1185792,1336203,1500488,1679475
%N A258402 a(n) = (n^2 + 4*n + 6) * n^2.
%C A258402 Consider a natural number r such that r has 15 proper divisors and 4 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 11. Where n is a positive integer, d(r^n) = (n^2 + 4*n + 6) * n^2.
%C A258402 The integers which satisfy the proper-divisor-prime-factor requirement are those of A033993.
%H A258402 Harvey P. Dale, <a href="/A258402/b258402.txt">Table of n, a(n) for n = 0..1000</a>
%H A258402 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A258402 From _Vincenzo Librandi_, Jun 06 2015: (Start)
%F A258402 G.f.: x*(11 + 17*x - 7*x^2 + 3*x^3)/(1 - x)^5.
%F A258402 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%F A258402 E.g.f.: exp(x)*x*(11 + 25*x + 10*x^2 + x^3). - _Stefano Spezia_, Oct 28 2023
%e A258402 The smallest integer which satisfies this is 210: It has 15 proper divisors (1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105) and 4 prime factors (2, 3, 5, 7), so d(210) = 11. The square of 210, 44100, we would expect to have a difference of 72 between the number of its proper divisors and prime factors, and with respectively 80 and 8, d(44100) = 72 indeed. Checking this with further integer powers of 210 will continue to generate terms in this sequence.
%t A258402 Table[(n^2 + 4n + 6) * n^2, {n, 0, 39}] (* _Alonso del Arte_, Jun 06 2015 *)
%t A258402 CoefficientList[Series[x (11 + 17 x - 7 x^2 + 3 x^3)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2015 *)
%t A258402 LinearRecurrence[{5,-10,10,-5,1},{0,11,72,243,608},40] (* _Harvey P. Dale_, May 05 2018 *)
%o A258402 (Magma) [(n^2+4*n+6)*n^2: n in [0..40]]; // _Vincenzo Librandi_, Jun 06 2015
%o A258402 (Magma) I:=[0,11, 72,243,608]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Jun 06 2015
%o A258402 (PARI) a(n)=(n^2+4*n+6)*n^2 \\ _Charles R Greathouse IV_, Jun 15 2015
%Y A258402 Cf. A033993.
%K A258402 nonn,easy
%O A258402 0,2
%A A258402 _Garrett Frandson_, Jun 05 2015
%E A258402 More terms from _Alonso del Arte_, Jun 06 2015