A164783 a(n) = 7^n-6.

1, 43, 337, 2395, 16801, 117643, 823537, 5764795, 40353601, 282475243, 1977326737, 13841287195, 96889010401, 678223072843, 4747561509937, 33232930569595, 232630513987201, 1628413597910443, 11398895185373137

Offset

1,2

Comments

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m (n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m (n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers.

References

Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem. Vol. 4, No. 2, Dec 1978, pp. 277-302.

Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.

D. Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.

D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.

Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.

Index entries for linear recurrences with constant coefficients, signature (8,-7).

Formula

a(n) = 7*a(n-1)+36 with n>1, a(1)=1. - Vincenzo Librandi, Nov 30 2010

G.f.: x*(1+35*x)/((1-x)*(1-7*x)). - Colin Barker, Mar 08 2012

a(n) = 8*a(n-1) - 7*a(n-2) for n>2, a(1)=1, a(2)=43. - Vincenzo Librandi, Feb 06 2013

a(n) = A000420(n) - 6 for n>0. - Michel Marcus, Aug 31 2013

Mathematica

CoefficientList[Series[(1 + 35 x)/((1-x) (1-7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[7 # + 36 & , 1, 18] (* Bruno Berselli, Feb 06 2013 *)
LinearRecurrence[{8, -7}, {1, 43}, 30] (* Harvey P. Dale, Nov 27 2014 *)

Prog

(Magma) [7^n-6: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013
(PARI) a(n)=7^n-6 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A000420.

Cf. A014232, A014224, A135535, A059266.

Sequence in context: A346575 A217621 A213554 * A291861 A142502 A272124

Adjacent sequences: A164780 A164781 A164782 * A164784 A164785 A164786

Keyword

nonn,easy

Author

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Extensions

More terms a(8)-a(19) from Vincenzo Librandi, Oct 29 2009

Status

approved