A014224 -id:A014224 - cp's OEIS frontend

A219050 Numbers k such that 3^k + 34 is prime.

1, 2, 3, 5, 7, 9, 10, 17, 27, 34, 51, 57, 61, 89, 98, 171, 547, 569, 769, 874, 1105, 2198, 2307, 3937, 4685, 5105, 5582, 11131, 11821, 15902, 24626, 36401, 46195, 50974, 65198, 66685

Offset: 1

Nicolas M. Perrault, Nov 10 2012

a(37) > 2*10^5. - Robert Price, Nov 24 2013

			For k = 2, 3^2 + 34 = 43 (prime), so 2 is in the sequence.

Cf. Sequences of numbers k such that 3^k + m is prime:
  (m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
  (m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
  (m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
  (m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
  (m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
  (m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Do[If[PrimeQ[3^n + 34], Print[n]], {n, 10000}]

is(n)=isprime(3^n+34) \\ Charles R Greathouse IV, Feb 17 2017

a(28)-a(36) from Robert Price, Nov 24 2013

A219051 Numbers k such that 3^k - 34 is prime.

Original entry on oeis.org

4, 7, 11, 13, 29, 32, 36, 44, 79, 157, 197, 341, 467, 996, 1421, 2479, 3269, 5203, 7987, 9341, 14836, 26047, 47816, 64304, 100693, 127597, 167167, 174697, 182089, 198791

Offset: 1

Nicolas M. Perrault, Nov 10 2012

a(31) > 2*10^5. - Robert Price, Nov 23 2013

			For k = 4, 3^4 - 34 = 47 and 47 is prime. Hence k = 4 is included in the sequence.

Cf. Sequences of numbers k such that 3^k + m is prime:
  (m = 2) A051783, (m = -2) A014224, (m = 4) A058958, (m = -4) A058959,
  (m = 8) A217136, (m = -8) A217135, (m = 10) A217137, (m = -10) A217347,
  (m = 14) A219035, (m = -14) A219038, (m = 16) A205647, (m = -16) A219039,
  (m = 20) A219040, (m = -20) A219041, (m = 22) A219042, (m = -22) A219043,
  (m = 26) A219044, (m = -26) A219045, (m = 28) A219046, (m = -28) A219047,
  (m = 32) A219048, (m = -32) A219049, (m = 34) A219050, (m = -34) A219051. Note that if m is a multiple of 3, 3^k + m is also a multiple of 3 (for k greater than 0), and as such isn't prime.

Do[If[PrimeQ[3^n - 34], Print[n]], {n, 1, 10000}]
Select[Range[10000], PrimeQ[3^# - 34] &] (* Alonso del Arte, Nov 10 2012 *)

is(n)=isprime(3^n-34) \\ Charles R Greathouse IV, Feb 17 2017

a(21)-a(30) from Robert Price, Nov 23 2013

A007593 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.

Original entry on oeis.org

21, 2133, 19521, 176661, 129127041, 328256967373616371221

Offset: 1

N. J. A. Sloane, David W. Wilson

67585198634817522935331173030319681 and 443426488243037769923934299701036035201 are also in the sequence, but their positions are unknown. - Jud McCranie, Dec 16 1999; updated by Max Alekseyev, Jun 03 2025
For all k in A014224, 3^(k-1)*(3^k-2) is in this sequence. - M. F. Hasler, Apr 25 2012
The known examples are all of the form 3^(k-1)*(3^k-2), where 3^k-2 is prime (cf. A014224). Conversely, from sigma(3^(k-1)*p)=(3^k-1)/2*(p+1) it is immediate that 2*sigma(n)=3n+1 for such numbers, i.e., they are 2-hyperperfect. (This is "form 3" with p=3 in McCranie's paper.) - M. F. Hasler, Apr 25 2012
Numbers k for which sigma(k) = (3k+1)/2, thus numbers k such that A000203(k) = A014682(k). Sequence A064989(a(n)), n >= 1, forms a subsequence of A337342. - Antti Karttunen, Aug 26 2020

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 21, p. 7, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
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, p. 561.
Daniel Minoli, Voice Over MPLS, McGraw-Hill, 2002, New York, NY, see pp. 112-134.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, PME 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.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 144.

Antal Bege, Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.
J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.
Daniel Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
Eric Weisstein's World of Mathematics, Hyperperfect Number

Cf. A000203, A000396, A014682, A028499, A028500, A028501, A028502, A034916, A220290, A337342.

Select[Range[2*10^5], #==2(DivisorSigma[1,#]-#-1)+1 &] (* Stefano Spezia, Sep 24 2024 *)

is(n)=n==2*(sigma(n)-n-1) + 1; \\ Charles R Greathouse IV, May 01 2013

a(6) from Jud McCranie confirmed and added by Max Alekseyev, Jun 03 2025

A128457 Numbers k such that 13^k - 2 is a prime.

Original entry on oeis.org

1, 2, 4, 5, 12, 78, 80, 90, 117, 120, 813, 1502, 2306, 2946, 6308, 13320, 26369, 31868, 44265, 81008

Offset: 1

Alexander Adamchuk, Mar 14 2007

13320 is a term found by Lelio R Paula 11/2006.
Numbers corresponding to a(13)..a(16) are probable primes. If n is of the form 4k+3 then 13^n-2 is composite, because 13^n-2 == (3^4)^k*3^3 - 2 == 25 == 0 (mod 5). So there is no term of the form 4k+3. - Farideh Firoozbakht, Dec 07 2009
a(21) > 2*10^5. - Robert Price, Oct 03 2014

Cf. A084714 (smallest prime of the form (2n-1)^k - 2).
Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Cf. A014224, A109080, A090669, A128455, A128458, A128459, A128460, A128461.

Do[ f = 13^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n,1,1000} ]

813 from Stefan Steinerberger, May 05 2007
a(12) from M. F. Hasler, Feb 07 2009
a(13)-a(16) from Farideh Firoozbakht, Dec 07 2009
a(17)-a(20) from Robert Price, Oct 03 2014

A128459 Numbers k such that 17^k - 2 is a prime.

Original entry on oeis.org

6, 24, 30, 106, 184, 232, 460, 1258, 3480, 5458, 32886

Offset: 1

Alexander Adamchuk, Mar 14 2007

No more terms through 50000. - Ryan Propper, Dec 06 2008

a(12) > 2*10^5. - Robert Price, Oct 23 2014

Cf. A084714 (smallest prime of the form (2n-1)^k - 2).
Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128460, A128461.

Do[ f = 17^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n,1,1000} ]

2 more terms from Stefan Steinerberger, May 05 2007
Two more terms from Ryan Propper, Jan 16 2008
One more term from Ryan Propper, Dec 06 2008

A128460 Numbers k such that 19^k - 2 is a prime.

Original entry on oeis.org

1, 2, 3, 13, 14, 19, 20, 23, 38, 1124, 7592, 11755, 12155, 12915, 14172, 15500, 20255, 28388, 184650

Offset: 1

Alexander Adamchuk, Mar 14 2007

No more terms through 50000. - Ryan Propper, Dec 04 2008

a(20) > 2*10^5. - Robert Price, Oct 22 2014

Cf. A084714 (smallest prime of the form (2n-1)^k - 2).
Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128461.

[n: n in [0..1000] | IsPrime(19^n-2)]; // Vincenzo Librandi, Oct 23 2014

Do[ f = 19^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n,1,1000} ]

1124 from Stefan Steinerberger, May 05 2007
7592 from Ryan Propper, Dec 31 2007
Additional terms from Ryan Propper, Dec 04 2008
a(19) from Robert Price, Oct 22 2014

A128455 Numbers k such that 9^k - 2 is a prime.

Original entry on oeis.org

1, 2, 3, 11, 45, 51, 260, 324, 390, 393, 1112, 3092, 4445, 10373, 15861, 18528, 97715, 112961

Offset: 1

Alexander Adamchuk, Mar 14 2007

a(19) > 2*10^5. - Robert Price, Aug 18 2014

Cf. A084714 (smallest prime of the form (2n-1)^k - 2).
Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k>(2n-1)).
Cf. A014224, A109080, A090669, A128457, A128458, A128459, A128460, A128461.

Do[ f = 9^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n,1,1000} ]

1112 from Stefan Steinerberger, May 05 2007
More terms from Ryan Propper, Jan 12 2008
a(15)-a(18) from Robert Price, Aug 18 2014

A128458 Numbers k such that 15^k - 2 is a prime.

Original entry on oeis.org

1, 2, 3, 7, 12, 17, 19, 51, 65, 550, 1460, 1641, 7035, 18002, 20963, 21163, 42563, 94906, 148048

Offset: 1

Alexander Adamchuk, Mar 14 2007

a(20) > 2*10^5. - Robert Price, Jun 23 2015

Cf. A084714 (smallest prime of the form (2n-1)^k - 2).
Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1)).
Cf. A014224, A109080, A090669, A128455, A128457, A128459, A128460, A128461.

Do[ f = 15^n - 2; If[ PrimeQ[ f ], Print[ {n, f} ] ], {n,1,1000} ]
Do[If[PrimeQ[15^n - 2], Print[n]], {n, 10^4}] (* Ryan Propper, Jun 06 2007 *)

550 from Stefan Steinerberger, May 05 2007
3 more terms from Ryan Propper, Jun 06 2007
a(14)-a(19) from Robert Price, Jun 23 2015

A128472 a(n) is the smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists.

Original entry on oeis.org

0, 79, 6103515623, 5764799, 31381059607

Offset: 1

Alexander Adamchuk, Mar 14 2007, Oct 01 2007

nonn

a(6) = 11^22420 - 2 was found by Rick L. Shepherd on Sep 29 2007. It has 23349 decimal digits and it is too large to include.

a(7) through a(12): {771936328432730777189183517369830159827426282764863750131729657829597399846468418688727, 98526125335693359373, 339448671314611904643504117119, 37589973457545958193355599, 1136272165922724266740722458520499, 480250763996501976790165756943039}.

Henri Lifchitz and Renaud Lifchitz: PRP Records. Probable Primes Top 10000.

Cf. A084714 (smallest prime of the form (2n-1)^k - 2).
Cf. A133856 (least number k > (2n-1) such that (2n-1)^k - 2 is prime).
Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.
Cf. A133858, A133982.

Edited by Max Alekseyev, Sep 18 2009

A014232 Primes of the form 3^k - 2.

Original entry on oeis.org

7, 79, 241, 727, 19681, 31381059607, 450283905890997361, 36472996377170786401, 8727963568087712425891397479476727340041447, 4638397686588101979328150167890591454318967698007

Offset: 1

Jud McCranie

nonn

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134) [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..20
Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645. [From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]

Cf. A000040, A007593, A014224 (corresponding k's).

[a: n in [1..200] | IsPrime(a) where a is 3^n-2]; // Vincenzo Librandi, Dec 07 2011

lst={};Do[p=3^n;If[PrimeQ[p-2],AppendTo[lst,p-2]],{n,2*5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 14 2010 *)
Select[3^Range[120]-2,PrimeQ] (* Harvey P. Dale, Aug 16 2011 *)

for(n=2,1e3,if(ispseudoprime(t=3^n-2),print1(n", "))) \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 3^A014224(n) - 2. - Elmo R. Oliveira, Nov 09 2023

cp's OEIS Frontend

A219050 Numbers k such that 3^k + 34 is prime.

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A219051 Numbers k such that 3^k - 34 is prime.

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A007593 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.

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A128457 Numbers k such that 13^k - 2 is a prime.

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A128459 Numbers k such that 17^k - 2 is a prime.

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A128460 Numbers k such that 19^k - 2 is a prime.

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A128455 Numbers k such that 9^k - 2 is a prime.

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A128458 Numbers k such that 15^k - 2 is a prime.

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