A305862 a(n) = 384*4^n - 576*3^n + 220*2^n - 14.

14, 234, 1826, 10770, 55154, 260274, 1167026, 5059890, 21442994, 89438514, 368866226, 1509026610, 6137242034, 24853275954, 100327829426, 404059098930, 1624486948274, 6522713868594, 26165182536626, 104883769004850, 420204307937714, 1682825158192434, 6737324873467826

Offset: 0

Author

Vincenzo Librandi, Jun 15 2018

Keywords

• nonn
• easy

Comments

From Bruno Berselli, Jun 15 2018: (Start)

a(0) = 2*7 and a(40) = 2*232110255958477539427146457 are semiprimes. For which values of n > 40 is a(n) semiprime?

For odd n, a(n) is divisible by 2*3.

For n == 3 (mod 4), a(n) is divisible by 2*3*5.

For n == 0 or 5 (mod 6), a(n) is divisible by 2*7.

For n == 2 or 4 (mod 5), a(n) is divisible by 2*11.

For n == 1 or 11 (mod 12), a(n) is divisible by 2*3*13.

For n == 15 (mod 16), a(n) is divisible by 2*3*5*17^2, etc.

If a(n) is divisible by 37 then it is also divisible by 3*5*7*13*19*73. (End)

Links

• Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018, page 11 (Lemma 3.4).
• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Crossrefs

Cf. A000453, A000918, A305861.

Programs

Magma

[384*4^n-576*3^n+220*2^n-14: n in [0..30]];

Mathematica

Table[384 4^n - 576 3^n + 220 2^n - 14, {n, 0, 30}]

PARI

a(n) = 384*4^n - 576*3^n + 220*2^n - 14; \\ Michel Marcus, Jul 03 2018

Formula

G.f.: 2*(7 + 47*x - 12*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)).

a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4).

a(n) = 14*A000453(n+4) + 94*A000453(n+3) - 24*A000453(n+2) for n>1. - Bruno Berselli, Jun 15 2018