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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000420.

Cf. A014232, A014224, A135535, A059266.

Programs

Magma

[7^n-6: n in [1..30]]; // Vincenzo Librandi, Feb 06 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 *)

PARI

a(n)=7^n-6 \\ Charles R Greathouse IV, Oct 07 2015

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

Extensions

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