A058959 -id:A058959 - 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

A063906 Numbers m such that m = 2*sigma(m)/3 - 1.

Original entry on oeis.org

15, 207, 1023, 2975, 19359, 147455, 1207359, 5017599, 2170814463, 58946212863, 1166333691001023, 36472996363223648799

Offset: 1

Jason Earls, Aug 30 2001

Original title: numbers n such that t(n) = s(n), where s(n) = sigma(n)-n-1 and t(n) = |s(n)-n|+1.
From Robert Israel, Jan 12 2016: (Start)
All terms are odd and satisfy A009194(m) = 1 or 3.
Includes 3^(k-1)*(3^k-4) for k in A058959. The first few terms of this form are 15, 207, 19359, 36472996363223648799.
Other terms include 3^15*43048567*1003302465131 = 619739816695811335405066239 and 3^15*43049011*808868950607 = 499643410492503517919703039. (End)
In other words, numbers m such that sigma(m)/(m+1) = 3/2. - Michel Marcus, Jan 03 2023
Some other terms: 374444425895728906239999, 2315893253834522244855807, 946345423297942718248771143999, 3181974057764759411641233725579002844163668627480799. - Max Alekseyev, Jul 30 2025

			sigma(1207359) = 1811040; 1811040 - 1207359 - 1 = 603680; abs(603680 - 1207359) + 1 = 603680.

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.

Cf. A000203, A009194, A014567, A058959.

for n := 1 to 4000000 do s := sigma(n) - n - 1; t := abs(s - n) + 1; if s = t then write(n," "); end; end;

[n: n in [1..6*10^6] | 2*DivisorSigma(1,n)/3-1 eq n]; // Vincenzo Librandi, Oct 10 2017

select(n -> numtheory:-sigma(n) = 3/2*(n+1), [seq(i,i=1..10^6,2)]); # Robert Israel, Jan 12 2016

Select[Range[10^6], 2 * DivisorSigma[1, #]/3 - 1 == # &] (* Giovanni Resta, Apr 14 2016 *)

s(n) = sigma(n)-n-1;
t(n) = abs(s(n)-n)+1;
for(n=1,10^8, if(t(n)==s(n),print1(n, ", ")))

More terms from Klaus Brockhaus, Sep 01 2001
a(9)-a(10) from Giovanni Resta, Apr 14 2016
Simpler title suggested by Giovanni Resta, Apr 14 2016, based on formula provided by Paolo P. Lava, Jan 12 2016
a(11)-a(12) from Max Alekseyev, Jul 30 2025

A165701 Numbers n such that 5^n-6 is prime.

Original entry on oeis.org

2, 4, 5, 6, 10, 53, 76, 82, 88, 242, 247, 473, 586, 966, 1015, 1297, 1825, 2413, 2599, 2833, 5850, 5965, 6052, 27199, 49704, 79000

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Oct 30 2009

Numbers corresponding to the a(n) for n>11 are probable prime.
If Q is a 4-perfect number and gcd(Q, 5*(5^a(n)-6))=1 then m=5^(a(n)-1)
(5^a(n)-6)*Q is a solution of the equation sigma(x)=5(x+Q)(see comment lines of the sequence A058959). 142990848 is the smallest 4-perfect number m such that 5 doesn't divide m.
a(27) > 10^5. - Robert Price, Feb 03 2014

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1
H. Lifchitz, R. Lifchitz: PRP Top Records Search for 5^n-6.

Cf. A007691, A054030, A058959.

Mathematica
```
Do[If[PrimeQ[5^n-6],Print[n]],{n,8888}]
```

is(n)=ispseudoprime(5^n-6) \\ Charles R Greathouse IV, Jun 13 2017

a(24)-a(26) from Robert Price, Feb 03 2014

A156555 Primes of the form 3^k - 4.

Original entry on oeis.org

5, 23, 239, 10460353199, 617673396283943, 450283905890997359, 36472996377170786399, 19383245667680019896796719, 67585198634817523235520443624317919, 1546132562196033993109383389296863818106322565999

Offset: 1

Vincenzo Librandi, Feb 10 2009

nonn

The next term, a(11), has 84 digits. - Harvey P. Dale, Jul 24 2011

			a(1) = 3^2 - 4 = 5 is the smallest prime of that form. - _M. F. Hasler_, Oct 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..17
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1
Henri & Renaud Lifchitz, PRP Records
OpenPFGW Project, Primality Tester

Cf. A000040, A058959 (corresponding k's).

Select[3^Range[200]-4,PrimeQ] (* Harvey P. Dale, Jul 24 2011 *)

for( k=2,999, is/*pseudo*/prime( p=3^k-4 ) & print1(p", ")) \\ M. F. Hasler, Oct 31 2009

a(n) = 3^A058959(n) - 4. - M. F. Hasler, Oct 31 2009

a(5) corrected by M. F. Hasler, Oct 31 2009

a(10) from Harvey P. Dale, Jul 24 2011

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A063906 Numbers m such that m = 2*sigma(m)/3 - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

ARIBAS

Magma

Maple

Mathematica

PARI

Extensions

A165701 Numbers n such that 5^n-6 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A156555 Primes of the form 3^k - 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI