A003307 -id:A003307 - cp's OEIS frontend

A120376 Primes of the form 2*5^k - 1.

1249, 31249, 305175781249, 119209289550781249, 1862645149230957031249, 111022302462515654042363166809082031249, 25243548967072377773175314089049159349542605923488736152648925781249

Offset: 1

Walter Kehowski, Jun 28 2006

nonn

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.

The next term has 125 digits. - Harvey P. Dale, Jan 26 2019

			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), A120375 (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), this sequence (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958.

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[2*5^Range[100]-1,PrimeQ] (* Harvey P. Dale, Jan 26 2019 *)

for(k=1, 1e3, if(ispseudoprime(p=2*5^k-1), print1(p, ", "))); \\ Altug Alkan, Sep 22 2018

a(n) = 2*5^A120375(n) - 1 = 2*5^(2*A002958(n)) - 1. - Jianing Song, Sep 22 2018

A272057 Numbers n such that 3*4^n - 1 is prime.

Original entry on oeis.org

1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859

Offset: 1

Tim Johannes Ohrtmann, Apr 19 2016

These are Williams primes to base 3.

Half of the even terms of A002235.

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

Cf. A002235, A000043, A003307, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907.

Select[Range[0, 100000], PrimeQ[3*4^# - 1] &]

for(n=1,10000, if(isprime(3*4^n-1), print1(n,", ")))

a(25) corrected and a(33)-a(36) added by Giovanni Resta, Apr 19 2016, using data from A002235.

A305237 Numbers m such that m, m+1 and m+2 all have primitive roots.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 17, 25, 81, 241

Offset: 1

Jianing Song, Jun 04 2018

nonn

Start of run of 3 consecutive numbers in A033948.
The next term is 3^541 - 2, which is too large to be included here. No more terms below 3^100000, or approximately 1.33*10^47712.
There is a multiple of 4 in every four consecutive positive integers and it clearly has no primitive roots if it is larger than 4. Again, there is a multiple of 3 in every three consecutive positive integers, so it must be a power of 3 or two times a power of 3, and the other two numbers must be odd prime powers or two times odd prime powers.
According to Pillai's conjecture, there're only finitely many solutions to |3^a - p^b| = 2, |3^a - 2*p^b| = 1, |p^a - 2*3^b| = 1 with a,b >= 2, p odd primes (no solution other than 3^3 - 5^2 = 2, 3^5 - 2*11^2 = 1 below 3^100000). So beyond (25, 26, 27) and (241, 242, 243), it's very likely that all three consecutive numbers with primitive roots are of the form (3^i, 3^i + 1, 3^i + 2), (3^j - 2, 3^j - 1, 3^j), (2*3^k - 1, 2*3^k, 2*3^k + 1) such that (3^i + 1)/2, 3^i + 2, 3^j - 2, (3^j - 1)/2, 2*3^k - 1, 2*3^k + 1 are primes, which only produces one more solution (3^541 - 2, 3^541 - 1, 3^541) below 3^1000000.

			81, 82, 83 all have primitive roots (in fact, their least common primitive root is 47), so 81 is a term.
Note that A014224 and A028491 have a term 541 in common, so 3^541 - 2, 3^541 - 1 and 3^541 all have primitive roots, so 3^541 - 2 is a term.

Jinyuan Wang, Table of n, a(n) for n = 1..11

Cf. A003306, A003307, A014224, A028491, A051783, A171381, A319450.

A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists.

Original entry on oeis.org

0, 6, 153, 66, 0, 3916, 0, 1770, 2556, 327645, 0, 1540, 0, 893862621, 8199225, 17766, 0, 76636, 0, 12720, 662976, 2096128, 0, 10296, 3357936, 416798777159765703, 6221628, 3611328, 0, 1734453, 0, 303810, 111576864636, 1420010137134674578503, 18051523357140153

Offset: 1

Jon E. Schoenfield, Dec 05 2018

nonn

The only primes p for which a(p) > 0 are those for which both 2*3^(p-1) - 1 and 2*3^(p-1) + 1 are prime: 2, 3, and any other primes p such that p-1 appears both in A003307 and A003306. (If such a prime p > 3 exists, then p exceeds 1360105.)

Conjecture: The only primes p for which a(p) > 0 are 2 and 3.

			For n=1, the only triangular number with exactly 2*1 = 2 divisors is T(2) = 2*(2+1)/2 = 3 (the only triangular number that is prime); thus, exists no pair of consecutive triangular numbers having exactly 2 divisors, so a(1)=0.
a(2) is 6 because T(3) = 3*(3+1)/2 = 6 and T(4) = 4*(4+1)/2 = 10 are the first two consecutive triangular numbers having exactly 2*2 = 4 divisors.

Cf. A000005, A000217, A063440, A081978, A319035.

Cf. A003306, A003307.

A177103 Numbers n >= 0 such that 2*3^n - 1 is not prime.

Original entry on oeis.org

0, 4, 5, 6, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Vincenzo Librandi

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003307 (Numbers n such that 2*3^n-1 is prime).

[ n: n in [0..100] | not IsPrime(2*3^n-1) ];

Select[Range[0, 100], !PrimeQ[2*3^# - 1] &] (* Vincenzo Librandi, Oct 15 2012 *)

A248849 Smallest k>0 such that 2^k*3^n-1 is a prime number.

Original entry on oeis.org

1, 1, 1, 3, 2, 5, 1, 1, 4, 3, 20, 1, 4, 13, 2, 11, 3, 101, 12, 1, 10, 9, 1, 11, 7, 27, 1, 347, 11, 73, 4, 7, 52, 93, 1, 7, 51, 73, 46, 11, 8, 41, 4, 51, 2, 5, 30, 11, 10, 3, 280, 11, 7, 17, 14, 1, 32, 11, 5, 11, 19, 1, 20, 17, 22, 133, 6, 1

Offset: 1

Pierre CAMI, Dec 03 2014

nonn

			2^1*3^1-1=5 prime so a(1)=1.
2^1*3^2-1=17 prime so a(2)=1.
2^1*3^3-1=53 prime so a(3)=1.

Pierre CAMI, Table of n, a(n) for n = 1..1500

Cf. A003307.

Flatten[{1,Table[k=0; While[Not[PrimeQ[2^k*3^n-1]],k++]; k,{n,2,100}]}] (* Vaclav Kotesovec, Dec 05 2014 *)

a(n)=1 for n=A003307(i).

A297348 Numbers k such that 12*13^k - 1 is prime.

Original entry on oeis.org

0, 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466

Offset: 1

Tim Johannes Ohrtmann, Feb 15 2018

Williams primes to base 12.

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

Cf. A003307, A272057, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907, A273523.

[n: n in [0..10000] |IsPrime(12*13^n-1)];

Select[Range[0, 10000], PrimeQ[12*13^n-1] &]

for(n=0, 10000, if(isprime(12*13^n-1), print1(n, ", ")))

a(16) from Michael S. Branicky, Sep 14 2022

a(17) from Michael S. Branicky, Nov 12 2024

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)))

A319535 Primes of the form 2*6^k - 1.

Original entry on oeis.org

11, 71, 431, 2591, 15551, 4353564671, 5642219814911, 341163456359156416511, 2046980738154938499071, 20628849596981071092343898111, 26734989077687468135677691953151, 207891275068097752223029732627709951, 269427092488254686881046533485512097791

Offset: 1

Jianing Song, Sep 22 2018

nonn

Primes in A164559.

Companion sequence of A057472. There are 49 terms known in this sequence.

			2*6^1 - 1 = 11, 2*6^2 - 1 = 71, 2*6^3 - 1 = 431, 2*6^4 - 1 = 2591 and 2*6^5 - 1 = 15551 are primes, but 2*6^6 - 1 = 93311 = 23*4057 is not.

K. D. Bajpai, Table of n, a(n) for n = 1..26

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), A120375 (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), this sequence (b=6), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958, A068231, A164559.

[k: n in [1..100] | IsPrime(k) where k is 2*6^n-1];  // K. D. Bajpai, Nov 15 2019

A319535:= n-> (2*6^n-1): select(isprime, [seq((A319535(n), n=1..200))]);  # K. D. Bajpai, Nov 15 2019

Select[Table[2*6^k-1,{k,1600}], PrimeQ[#]&]  (* K. D. Bajpai, Nov 15 2019 *)

for(n=1, 99, my(t); if(ispseudoprime(t=2*6^n-1), print1(t", ")))

a(n) = 2*6^A057472(n) - 1.

A107651 Numbers n such that phi(sigma(n)) + phi(phi(n)) = n.

Original entry on oeis.org

3, 28, 108, 2352, 2544, 7936, 13632, 26736, 209904, 256608, 1394112, 2052864, 2169456, 2490864, 11942400, 18884416, 258072480, 415272960, 2064579840, 3737456640, 3963371520, 4672512000

Offset: 1

Farideh Firoozbakht, May 26 2005

nonn

If both (3^n-1)/2 and 2*3^n-1 are prime then 48*(2*3^n-1) is in the sequence (the proof is easy). So if n is in the intersection of A028491 and A003307 then 48*(2*3^n-1) is in this sequence. Conjecture: There exist only two such terms, namely 2544 and 209904.

If both (3^n*31-1)/2 and 2*3^n*31-1 are prime then 48*(2*3^n*31-1) is in the sequence (the proof is easy). Conjecture: There exist only three such terms, namely 26736, 2169456, and 26376103844085843261484656.

			18884416 is in the sequence because phi(sigma(18884416)) + phi(phi(18884416)) = 18884416.

Cf. A003307, A028491, A107652, A107653.

Do[If[n == EulerPhi[DivisorSigma[1, n]] + EulerPhi[EulerPhi[n]], Print[n] ], {n, 10000000}]

is(n)=eulerphi(sigma(n))+eulerphi(eulerphi(n))==n \\ Charles R Greathouse IV, Mar 06 2013

a(17)-a(22) from Donovan Johnson, Mar 06 2013

cp's OEIS Frontend

A120376 Primes of the form 2*5^k - 1.

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A272057 Numbers n such that 3*4^n - 1 is prime.

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A305237 Numbers m such that m, m+1 and m+2 all have primitive roots.

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A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists.

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A177103 Numbers n >= 0 such that 2*3^n - 1 is not prime.

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Magma

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A248849 Smallest k>0 such that 2^k*3^n-1 is a prime number.

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Mathematica

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A297348 Numbers k such that 12*13^k - 1 is prime.

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Magma

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

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A319535 Primes of the form 2*6^k - 1.