A135535 -id:A135535 - cp's OEIS frontend

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

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

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.

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)

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).

Cf. A000420.

Cf. A014232, A014224, A135535, A059266.

[7^n-6: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

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 *)

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

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

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

A164784 a(n) = 6^n-5.

Original entry on oeis.org

1, 31, 211, 1291, 7771, 46651, 279931, 1679611, 10077691, 60466171, 362797051, 2176782331, 13060694011, 78364164091, 470184984571, 2821109907451, 16926659444731, 101559956668411, 609359740010491, 3656158440062971

Offset: 1

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

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. For additional references, beyond key ones listed below, see A164783.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, 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 (7, -6).

[6^n-5: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

CoefficientList[Series[(1 + 24 x)/(1 - 7 x + 6 x^2), {x, 0, 30}],x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 6*a(n-1)+25 with n>1, a(1)=1. - Vincenzo Librandi, Oct 29 2009
G.f.: x*(1 + 24*x)/(1 - 7*x + 6*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 4 + (exp(5*x) - 5)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

A164785 a(n) = 5^n - 4.

Original entry on oeis.org

1, 21, 121, 621, 3121, 15621, 78121, 390621, 1953121, 9765621, 48828121, 244140621, 1220703121, 6103515621, 30517578121, 152587890621, 762939453121, 3814697265621, 19073486328121, 95367431640621, 476837158203121

Offset: 1

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

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. For additional references, beyond key ones listed below, see A164783.

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A059613.

[5^n-4: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

5^Range[50]-4 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{1,21},30] (* or *) NestList[5 # + 16 &, 1, 30] (* Harvey P. Dale, Jun 07 2012 *)
CoefficientList[Series[(1 + 15 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 5*a(n-1) + 16 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 30 2010
a(n) = 6*a(n-1) - 5*a(n-2); a(1)=1, a(2)=21. - Harvey P. Dale, Jun 07 2012
G.f.: x*(1 + 15*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 3 + (exp(4*x) - 4)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(9)-a(21) from Vincenzo Librandi, Oct 29 2009

A164786 a(n) = 8^n-7.

Original entry on oeis.org

1, 57, 505, 4089, 32761, 262137, 2097145, 16777209, 134217721, 1073741817, 8589934585, 68719476729, 549755813881, 4398046511097, 35184372088825, 281474976710649, 2251799813685241, 18014398509481977, 144115188075855865

Offset: 1

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

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. For additional references, beyond key ones listed below, see A164783.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, 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 (9, -8).

[8^n-7: n in [1..20]]; // Vincenzo Librandi, Aug 22 2011

8^Range[20]-7 (* or *) LinearRecurrence[{9,-8},{1,57},20] (* Harvey P. Dale, Jan 24 2013 *)

a(n) = 8*a(n-1)+49, with a(1)=1. - Vincenzo Librandi, Nov 30 2010
G.f.: x*(1+48*x)/(1-9*x+8*x^2). a(n) = 9*a(n-1)-8*a(n-2). - Colin Barker, Jan 28 2012
E.g.f.: 6 + (exp(7*x) - 7)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(7)-a(19) from Vincenzo Librandi, Oct 29 2009

A383872 Nonprime numbers whose sum of proper divisors is a power of 4.

Original entry on oeis.org

9, 12, 26, 56, 76, 122, 332, 992, 2042, 3344, 4336, 8186, 16256, 32762, 227744, 266176, 269072, 299576, 856544, 2097146, 5385812, 8388602, 16580864, 17895664, 19173944, 33554426, 61008020, 67100672, 201931760, 1074789376, 1108378592, 17179738112, 62472251540, 68700578816

Offset: 1

Hans Ulrich Keller, May 13 2025

nonn

Includes 2*p for p in A135535. - Robert Israel, May 13 2025
From David A. Corneth, May 13 2025: (Start)
If a(n) = m*p where p is the largest prime divisor and has multiplicity 1 and s = sigma(m) then p = (4^k - s) / (s - m). Using this, a(44), a(45) and a(46) are at most 4971704751572, 44088037271892 and 44358570286896 respectively.
If a(n) is odd then a(n) is a perfect square. Proof: Suppose a(n) is not a perfect square. Then sigma(a(n)) is even and so sigma(a(n)) - a(n) = 4^k. As sigma(a(n)) - a(n) = 4^k then sigma(a(n)) - a(n) = 1. As a(n) is composite this has no solutions. (End)
For the first 43 terms, a(1) is the only square term and for the other terms, a(n) has a squarefree odd part. However, this is not always true as 44088037271892 (see above) is a term and its odd part is not squarefree. - Chai Wah Wu, May 19 2025

			12 is not prime; 12 has proper divisors 1, 2, 3, 4, and 6, with a sum of 16. This is a square number as well as a power of 2.

Amiram Eldar, Table of n, a(n) for n = 1..43 (calculated from the b-file at A279731)

Intersection of A048699 and A279731.

Cf. A001065, A135535.

filter:= proc(n) local s;
  s:= numtheory:-sigma(n)-n;
  s > 1 and s = 4^padic:-ordp(s,4)
end proc:
select(filter, [$4..10^7]); # Robert Israel, May 13 2025

Zweierpotenzen = {};
Quadratzahlen = {};
Beides = {};
For[k = 1, k <= 50000000, k++,
  SumET = Total[Divisors[k]] - k;
  If[IntegerQ[Log[2, SumET]] && PrimeQ[k] == False,
   AppendTo[Zweierpotenzen, k]];
  If[IntegerQ[Sqrt[SumET]] && PrimeQ[k] == False,
   AppendTo[Quadratzahlen, k ]]];
Beides = Intersection[Zweierpotenzen, Quadratzahlen];
Beides

isok(k) = if (!ispseudoprime(k), my(s=sigma(k)-k, z); issquare(s) && (ispower(s, , &z) && (z==2))); \\ Michel Marcus, May 13 2025

a(27)-a(29) from Michel Marcus, May 13 2025

a(30)-a(34) from Amiram Eldar, May 13 2025

A181285 Primes of the form 5^k - 4.

Original entry on oeis.org

3121, 78121, 30517578121, 710542735760100185871124267578121, 413590306276513837435704346034981426782906055450439453121

Offset: 1

Jonathan D. B. Hodgson, Oct 12 2010

nonn

			3121 = 5^5 - 4 is prime and therefore is in the sequence.

Cf. A059613, A135535, A228028.

n:=1000: S:={}: for i from 1 to n do if type(5^i-4,prime)=true then S:=S union {5^i-4} end if od; S;

Select[5^Range[90]-4,PrimeQ] (* Harvey P. Dale, Aug 23 2013 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A164784 a(n) = 6^n-5.

Views

Author

Keywords

Comments

References

Links

Programs

Magma

Mathematica

Formula

A164785 a(n) = 5^n - 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A164786 a(n) = 8^n-7.

Views

Author

Keywords

Comments

References

Links

Programs

Magma

Mathematica

Formula

Extensions

A383872 Nonprime numbers whose sum of proper divisors is a power of 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A181285 Primes of the form 5^k - 4.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica