A003307 -id:A003307 - cp's OEIS frontend

A005541 Numbers k such that 8*3^k - 1 is prime.

0, 1, 2, 4, 10, 17, 50, 170, 184, 194, 209, 641, 1298, 4034, 5956, 7154, 9970, 35956, 42730, 132004, 190610

Offset: 1

a(22) > 2*10^5. - Robert Price, Mar 16 2014
All terms are verified primes (i.e., not probable primes). - Robert Price, Mar 16 2014
896701 is a term, found in 2010 (see link). - Jeppe Stig Nielsen, Jul 31 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

The Prime Pages, 8*3^896701 - 1.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A003307, A002235, A046865, A079906, A046866, A001771, A056725, A046867, A079907.

lst={};Do[If[PrimeQ[8*3^n-1], AppendTo[lst, n]], {n, 13^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)

is(n)=isprime(8*3^n-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Douglas Burke (dburke(AT)nevada.edu)
0 prepended by Vincenzo Librandi, Sep 26 2012
a(18)-a(21) from Robert Price, Mar 16 2014

A163667 Numbers n such that sigma(n) = 9*phi(n).

Original entry on oeis.org

30, 264, 714, 3080, 3828, 6678, 10098, 12648, 21318, 22152, 24882, 44660, 49938, 61344, 86304, 94944, 118296, 129504, 130356, 147560, 183396, 199386, 201756, 207264, 216936, 248710, 258440, 265914, 275196, 290290, 321204, 505164, 628776, 706266, 706836

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Aug 09 2009

This sequence is a subsequence of A011257 because sqrt(phi(n)*sigma(n)) = 3*phi(n).

If 2^p-1 and 2*3^k-1 are two primes greater than 5 then n = 2^(p-2)*(2^p-1)*3^(k-1)*(2*3^k-1) (the product of two relatively prime terms 2^(p-2)*(2^p-1) and 3^(k-1)*(2*3^k-1) of A011257) is in the sequence. The proof is easy.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Cf. A000010, A000043, A000203, A000668, A003307, A011257, A079363.

Select[Range[700000],DivisorSigma[1,# ]==9EulerPhi[ # ]&]

is(n)=sigma(n)==9*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A111974 Primes of the form 2*3^k + 1.

Original entry on oeis.org

3, 7, 19, 163, 487, 1459, 39367, 86093443, 258280327, 411782264189299, 116299474006080119380780339, 3140085798164163223281069127, 84782316550432407028588866403, 20602102921755074907947094535687, 1910009901593650473786381403548828023839870277948686259673707683

Offset: 1

T. D. Noe, Aug 24 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..23
Hartosh Singh Bal and Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019-2020.

Cf. A003306 (k such that 2*3^k + 1 is prime), A003307 (k such that 2*3^k - 1 is prime), A052919.

Mathematica
```
Select[2*3^Range[100]+1, PrimeQ]
```

a(n) = A052919(A003306(n)+1). - Amiram Eldar, Jul 18 2025

a(15) from Amiram Eldar, Jul 18 2025

A120378 Integers n such that 2*11^n-1 is prime.

Original entry on oeis.org

2, 8, 248, 2474, 2900, 6600, 24746, 105704

Offset: 1

Walter Kehowski, Jun 28 2006

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 241 is 181 in base 12.

a(9) > 2*10^5. - Robert Price, Nov 06 2015

			a(1)=2 since 2*11^2-1=241 is the first prime of this form.

Cf. A000043, A000668, A002957, A002959, A003307, A079363, A055558.

for w to 1 do for k from 1 to 2000 do n:=2*11^k-1; if isprime(n) then printf("%d, %d",k,n) fi od od;

Select[Range[0, 200000], PrimeQ[2*11^# - 1] &] (* Robert Price, Nov 06 2015 *)

a(n) = n-th integer k such that 2*11^k-1 is prime.

More terms from Ryan Propper, Jan 14 2008

a(7)-a(8) from Robert Price, Nov 06 2015

A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

Original entry on oeis.org

5, 8, 13, 11, 18, 25, 14, 23, 32, 41, 17, 28, 39, 50, 61, 20, 33, 46, 59, 72, 85, 23, 38, 53, 68, 83, 98, 113, 26, 43, 60, 77, 94, 111, 128, 145, 29, 48, 67, 86, 105, 124, 143, 162, 181, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221, 35, 58, 81, 104, 127, 150, 173, 196, 219, 242, 265

Offset: 1

Vincenzo Librandi, Jan 13 2009

First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   5;
   8, 13;
  11, 18, 25;
  14, 23, 32, 41;
  17, 28, 39, 50,  61;
  20, 33, 46, 59,  72,  85;
  23, 38, 53, 68,  83,  98, 113;
  26, 43, 60, 77,  94, 111, 128, 145;
  29, 48, 67, 86, 105, 124, 143, 162, 181;
  32, 53, 74, 95, 116, 137, 158, 179, 200, 221; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.

Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).

[2*n*k + n + k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 14 2023

Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - R. J. Mathar, Jan 15 2009, Jan 05 2011
From G. C. Greubel, Oct 14 2023: (Start)
T(n, n) = A001844(n).
T(n, n-1) = A001105(n), n >= 2.
T(n, n-2) = A142463(n-1), n >= 3.
T(n, n-3) = (-1)*A147973(n+2), n >= 4.
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).
G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)

A120375 Integers k such that 2*5^k - 1 is prime.

Original entry on oeis.org

4, 6, 16, 24, 30, 54, 96, 178, 274, 1332, 2766, 3060, 4204, 17736, 190062, 223536, 260400, 683080

Offset: 1

Walter Kehowski, Jun 28 2006

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 1249 is 881 in base 12.

a(16) > 2*10^5. - Robert Price, Mar 14 2015

			a(1) = 4 since 2*5^4 - 1 = 1249 is the first prime.

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), this sequence (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).
Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), A120376 (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958.

[n: n in [0..2800] |IsPrime(2*5^n - 1)]; // Vincenzo Librandi, Sep 23 2018

for w to 1 do for k from 1 to 2000 do n:=2*5^k-1; if isprime(n) then printf("%d, %d ",k,n) fi od od;

Select[Range[0, 100], PrimeQ[2*5^# - 1] &] (* Robert Price, Mar 14 2015 *)

isok(k) = ispseudoprime(2*5^k-1); \\ Altug Alkan, Sep 22 2018

a(n) = 2*A002958(n).

More terms from Ryan Propper, Mar 28 2007
a(14) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 02 2007
a(15) from Robert Price, Mar 14 2015
a(16)-a(18) from Jorge Coveiro and Tyler NeSmith, Jun 14 2020

A120377 Primes of the form 2*11^k-1.

Original entry on oeis.org

241, 428717761

Offset: 1

Walter Kehowski, Jun 28 2006

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 241 is 181 in base 12.

The values of k < 1000 that yield primes are 2, 8, 248. - T. D. Noe, Nov 16 2006

			a(1) = 241 since 2*11^2-1 = 241 is the first prime.

Amiram Eldar, Table of n, a(n) for n = 1..3

Cf. A000043, A000668, A002957, A002959, A003307, A055558, A079363, A120378.

for w to 1 do for k from 1 to 2000 do n:=2*11^k-1; if isprime(n) then printf("%d, %d",k,n) fi od od;

Select[2*11^Range[1000]-1, PrimeQ] (* T. D. Noe, Nov 16 2006 *)

a(n) = n-th number such that 2*11^k-1 that is prime for some k.

a(n) = 2*11^A120378(n)-1. - R. J. Mathar, Mar 06 2010

Corrected by T. D. Noe, Nov 16 2006

A268061 Numbers k such that 7*8^k - 1 is prime.

Original entry on oeis.org

3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299

Offset: 1

Robert Price, Jan 25 2016

a(10) > 2*10^5.

Terms are A001771(n)/3 that are integers.

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. similar sequences of the form k*(k+1)^n-1: A003307 (k=2), ... (k=3), A046865 (k=4), A079906 (k=5), A046866 (k=6), this sequence (k=7), ... (k=8), A056725 (k=9), A046867 (k=10), A079907 (k=11).

Cf. A001771, A002235, A005541, A247260.

Select[Range[0, 200000], PrimeQ[7*8^# - 1] &]

lista(nn) = for(n=1, nn, if(ispseudoprime(7*8^n-1), print1(n, ", "))) \\ Altug Alkan, Jan 25 2016

A119591 Least k such that 2*n^k - 1 is prime.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 1, 10, 1, 1, 6, 1, 2, 6, 1, 2, 136, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 2, 4, 4, 1, 2, 1, 1, 44, 1, 1, 2, 1, 3, 2, 5, 3, 2, 2, 1, 4, 1, 768, 4, 1, 1, 52, 34, 2, 132, 1, 1, 14, 7, 1, 2, 2, 1, 8, 1, 2, 10, 1, 24, 60, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1

Offset: 2

Pierre CAMI, Jun 01 2006

From Eric Chen, Jun 01 2015: (Start)
Conjecture: a(n) is defined for all n.
a(303) > 10000, a(304)..a(360) = {1, 2, 11, 1, 990, 1, 1, 2, 2, 4, 74, 5, 1, 10, 6, 6, 4, 1, 1, 2, 1, 9, 12, 1, 80, 2, 1, 1, 2, 14, 3, 2, 3, 1, 12, 1, 60, 36, 1, 8, 4, 34, 1, 522, 3, 15, 14, 1, 6, 2, 3, 1, 4, 5, 4, 10, 1}.
a(n) = 1 if and only if n is in A006254. (End)
From Eric Chen, Sep 16 2021: (Start)
Now a(303) is known to be 40174, also other terms > 10000: a(383) = 20956, a(515) = 58466, a(522) = 62288, a(578) = 129468, a(581) > 400000, a(590) = 15526, a(647) = 21576, a(662) = 16590, a(698) = 127558, a(704) = 62034, see the a-file and the references.
a(n) = 2 if and only if n is in A066049 but not in A006254.
a(n) = 3 if and only if n is in A214289 but not in A006254 or A066049. (End)

Eric Chen, Table of n, a(n) for n = 2..580
Gary Barnes, Riesel conjectures and proofs
Eric Chen, Table of n, a(n) for n = 2..2050 status
Prime Wiki, Riesel prime small bases least n

Cf. A119624, A253178.

Numbers r such that 2*k^r-1 is prime: A090748 (k=2), A003307 (k=3), A146768 (k=4), A120375 (k=5), A057472 (k=6), A002959 (k=7), ... (k=8), ... (k=9), A002957 (k=10), A120378 (k=11), ... (k=12), A174153 (k=13), A273517 (k=14), ... (k=15), ... (k=16), A193177 (k=17), A002958 (k=25).

f[n_] := Block[{k = 0}, While[ ! PrimeQ[2*n^k - 1], k++ ]; k ]; Table[f[n], {n, 2, 106}] (* Ray Chandler, Jun 08 2006 *)

a(n) = for(k=1, 2^24, if(ispseudoprime(2*n^k-1), return(k))) \\ Eric Chen, Jun 01 2015

From Eric Chen, Sep 16 2021: (Start)
a(6*n) = A098873(n).
a(2^n) = A279095(n).
a(A006254(n)) = 1.
a(A066049(n)) <= 2.
a(A214289(n)) <= 3. (End)

Corrected and extended by Ray Chandler, Jun 08 2006

A245241 Integers n such that 6 * 7^n + 1 is prime.

Original entry on oeis.org

0, 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572

Offset: 1

Robert Price, Nov 14 2014

All terms correspond to verified primes, that is, not merely probable primes.

a(14) > 2*10^5.

			4 is in this sequence because 6 * 7^4 + 1 = 14407, which is prime.

Cf. A003307, A002235, A046865, A079906, A001771, A005541, A056725, A046867, A079907.

Cf. A005537, A005538, A005539, A216889, A216890, A247260.

Select[Range[0,200000], PrimeQ[6 * 7^# + 1] &]

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A005541 Numbers k such that 8*3^k - 1 is prime.

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A163667 Numbers n such that sigma(n) = 9*phi(n).

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A111974 Primes of the form 2*3^k + 1.

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A120378 Integers n such that 2*11^n-1 is prime.

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A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

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A120375 Integers k such that 2*5^k - 1 is prime.

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A120377 Primes of the form 2*11^k-1.

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