A050524 -id:A050524 - cp's OEIS frontend

A052996 Expansion of (1+x^2-x^3)/((1-x)(1-2x)).

1, 3, 8, 17, 35, 71, 143, 287, 575, 1151, 2303, 4607, 9215, 18431, 36863, 73727, 147455, 294911, 589823, 1179647, 2359295, 4718591, 9437183, 18874367, 37748735, 75497471, 150994943, 301989887, 603979775, 1207959551, 2415919103

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-4, 14.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1074
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A050524 (primes of this sequence).

[Floor(9*2^(n-2) - 1): n in [0..40]]; // Vincenzo Librandi, Sep 24 2011

spec := [S, {S=Prod(Union(Prod(Z,Z), Sequence(Z)), Sequence(Union(Z,Z)))}, unlabeled ]: seq(combstruct[count](spec, size=n), n=0..20);

a[0] := 1; a[1] := 3; a[2] := 8; a[n_] := 2*a[n - 1] + 1; Table[a[n], {n, 0, 12}] (* L. Edson Jeffery, Dec 18 2014 *)
CoefficientList[ Series[(1 + x^2 - x^3)/((1 - x) (1 - 2 x)), {x, 0, 30}], x] (* Robert G. Wilson v, Jul 29 2015 *)
LinearRecurrence[{3,-2},{1,3,8,17},40] (* Harvey P. Dale, Feb 11 2018 *)

Vec((1+x^2-x^3)/((1-x)*(1-2*x)) + O(x^50)) \\ Michel Marcus, Jul 30 2015

Recurrence: {-2*a(n)+a(n+1)-1=0, a(0)=1, a(1)=3, a(2)=8}.

a(n) = 9*2^(n-2) - 1 for n > 1. - Brad Clardy, Sep 23 2011

More terms from James Sellers, Jun 06 2000

A002236 Numbers k such that 9*2^k - 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 15, 21, 43, 63, 99, 109, 159, 211, 309, 343, 415, 469, 781, 871, 939, 1551, 3115, 3349, 5589, 5815, 5893, 7939, 8007, 11547, 12495, 22555, 23647, 35647, 83415, 103059, 184999, 275859, 384243, 484975, 503893, 828709, 1010277, 1419855, 1481821

Offset: 1

N. J. A. Sloane

Even exponents can give at most semiprimes (see A181490). - Jeppe Stig Nielsen, Jun 08 2023

H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..48
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
C. K. Caldwell, The Prime Pages
The Prime Pages, Search output for 9*2^k - 1
Prime-Wiki, Riesel primes of the form 9*2^n - 1
Kosmaj, Riesel list k<300.
H. Riesel, Lucasian criteria for the primality of N=h.2^n-1, Math. Comp., 23 (1969), 869-875.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050524.

b=9; i=0; Table[While[i++; cp=b*2^i-1; !PrimeQ[cp]]; i, {j, 1, 22}] (* Lei Zhou, Nov 08 2013 *)
Select[Range[3400],PrimeQ[9*2^#-1]&] (* The program generates the first 22 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Harvey P. Dale, Sep 01 2020 *)

is(n)=ispseudoprime(9*2^n-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008

a(42)-a(43) communicated by Jeppe Stig Nielsen, Jun 08 2023

A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

Original entry on oeis.org

34, 568, 147328, 603971584, 9663643648, 39582416502784, 696341272098017608537735168, 765635325572111542783369494684623699968, 3615610599582728119969414707766982030374842621310535527825408, 3791242500068058721125048996612134914443116117566314438843154038784

Offset: 1

Jinyuan Wang, Feb 12 2020

nonn

This is the case h = 8 of the h-perfect numbers as defined in the Harborth link.

			34 is a term since sigma(34) + sigma(8*34) = 612, that is 18*34.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A002236, A050524, A136539, A281993, A330413.

isok(m) = sigma(m) + sigma(8*m) == 18*m;

a(n) = 2^A002236(n) * A050524(n).

cp's OEIS Frontend

A052996 Expansion of (1+x^2-x^3)/((1-x)(1-2x)).

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A002236 Numbers k such that 9*2^k - 1 is prime.

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A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

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cp's OEIS Frontend

A052996 Expansion of (1+x^2-x^3)/((1-x)*(1-2*x)).

Views

Author

Keywords

Links

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Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A002236 Numbers k such that 9*2^k - 1 is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A330412 Integers m such that sigma(m) + sigma(8*m) = 18*m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A052996 Expansion of (1+x^2-x^3)/((1-x)(1-2x)).

A330412 Integers m such that sigma(m) + sigma(8m) = 18m.