A258618 a(n) = (4*n+9)*n^2.

0, 13, 68, 189, 400, 725, 1188, 1813, 2624, 3645, 4900, 6413, 8208, 10309, 12740, 15525, 18688, 22253, 26244, 30685, 35600, 41013, 46948, 53429, 60480, 68125, 76388, 85293, 94864, 105125, 116100, 127813, 140288, 153549, 167620, 182525, 198288, 214933

Offset: 0

Author

Garrett Frandson, Jun 05 2015

Keywords

• nonn
• easy

Comments

Consider a natural number r such that r has 19 proper divisors and 6 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 13. Where n is a positive integer, d(r^n)=(4*n+9)*n^2.

The integers that satisfy the proper-divisor-prime-factor requirement are those of A179644.

Examples

			The smallest integer that satisfies this is 240: It has 19 proper divisors (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120) and 6 prime factors (2, 2, 2, 2, 3, 5), so d(240)=13. The square of 240, 57600, we would expect to have a difference of 68 between the number of its proper divisors and prime factors, and with respectively 80 and 12, d(57600)=68 indeed. Checking this with further integer powers of 240 will continue to generate terms in this sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A179644.

Programs

Magma

[(4*n+9)*n^2: n in [0..40]]; // Vincenzo Librandi, Jun 06 2015

Mathematica

Table[(4 n + 9) n^2, {n, 0, 40}] (* Vincenzo Librandi, Jun 06 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,13,68,189},40] (* Harvey P. Dale, Sep 12 2020 *)

PARI

vector(50,n,n--;(4*n+9)*n^2) \\ Derek Orr, Jun 21 2015

Formula

G.f.: x*(13+16*x-5*x^2)/(1-x)^4. - Vincenzo Librandi, Jun 06 2015

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Jun 06 2015

Extensions

More terms from Vincenzo Librandi, Jun 06 2015