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A258617
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a(n) = (4*n+8)*n^2.
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0
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0, 12, 64, 180, 384, 700, 1152, 1764, 2560, 3564, 4800, 6292, 8064, 10140, 12544, 15300, 18432, 21964, 25920, 30324, 35200, 40572, 46464, 52900, 59904, 67500, 75712, 84564, 94080, 104284, 115200, 126852, 139264, 152460, 166464, 181300, 196992, 213564
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OFFSET
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0,2
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COMMENTS
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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.
The integers that satisfy the proper-divisor-prime-factor requirement are those of A179643.
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LINKS
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Table of n, a(n) for n=0..37.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = 4*A152618(n+1).
G.f.: 4*x*(3+4*x-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
For any m, let x=A179643(m), then a(n) = A000005(x^n) - A001222(x^n). - Michel Marcus, Jul 09 2015
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EXAMPLE
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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).
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.
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MATHEMATICA
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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 *)
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PROG
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(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
(PARI) vector(50, n, n--; (4*n+8)*n^2) \\ Derek Orr, Jun 21 2015
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CROSSREFS
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Cf. A152618, A179643.
Sequence in context: A154507 A105916 A193872 * A307061 A074359 A104062
Adjacent sequences: A258614 A258615 A258616 * A258618 A258619 A258620
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KEYWORD
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nonn,easy
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AUTHOR
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Garrett Frandson, Jun 05 2015
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EXTENSIONS
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More terms from Vincenzo Librandi, Jun 06 2015
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STATUS
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approved
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