A004062 -id:A004062 - cp's OEIS frontend

A185073 Numbers n such that (34^n - 1)/33 is prime.

13, 1493, 5851, 6379, 125101

Offset: 1

Robert Price, Mar 10 2012

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Lifchitz, Mersenne and Fermat primes field
P. Bourdelais, A Generalized Repunit Conjecture

Cf. A028491, A004061, A004062, A004063, A004064, A004023, A005808, A016054, A006032, A006033, A006034, A006035, A098438, A127995-A128005.

Select[Prime[Range[100]], PrimeQ[(34^#-1)/33]&]

isok(n) = isprime((34^n-1)/33); \\ Michel Marcus, Mar 13 2016

lista(nn) = for(n=1, nn, if(ispseudoprime((34^n - 1)/33), print1(n, ", "))); \\ Altug Alkan, Mar 13 2016

a(5)=125101 corresponds to a probable prime discovered by Paul Bourdelais, Nov 20 2017

A198725 Primes of the form (6^n-11)/5.

Original entry on oeis.org

5, 41, 257, 1553, 15672832817, 121871948002097, 4387390128075569, 161656255492952812128627920091307258673, 34917751186477807419783630739722367873841

Offset: 1

Gilbert Mozzo, Oct 29 2011

nonn

These primes are also given by sum 6^k -1 with k>0 and are then companions of A165210 which corresponds also to sum 6^k +1 with k>0. (Be careful: there is a shifting between the k and the n values).

Corresponding exponents n are in A199165. - Gilbert Mozzo, Nov 05 2011

			(6^4-11)/5=257, which is in the sequence because it is prime.

Cf. A165210, A004062, A199165.

[(6^n-11)/5: n in [1..10^3] | IsPrime((6^n-11) div 5)];

lst={}; Do[If[PrimeQ[(6^n-11)/5], Print[(6^n-11)/5]; AppendTo[lst, (6^n-11)/5]], {n, 10^6}];

for(n=1,1e4,if(ispseudoprime(t=6^n\5-2),print1(t", "))) \\ Charles R Greathouse IV, Nov 01 2011

A199165 Numbers n such that (6^n-11)/5 is prime.

Original entry on oeis.org

2, 3, 4, 5, 14, 19, 21, 50, 53, 136, 146, 1255, 1448, 1839, 2053, 2496, 4060, 5041, 8410, 14090, 14940, 19759, 29871, 44836, 78175, 114398, 120946, 137845, 461108, 727496, 840316

Offset: 1

Gilbert Mozzo, Nov 03 2011

			a(4) = 5  because  (6^5-11)/5 = 1553  is prime.

Henri & Renaud Lifchitz, PRP Records.

Cf. A004062, A165210, A198725.

lst={}; Do[If[PrimeQ[(6^n-11)/5], Print[n]; AppendTo[lst, n]], {n, 10^6}]; lst

is(n)=ispseudoprime((6^n-11)/5) \\ Charles R Greathouse IV, Jun 13 2017

a(23)-a(28) are probable primes discovered by Paul Bourdelais, Nov 15 2011
a(23)-a(28) independently confirmed as probable primes using Mathematica PrimeQ function by Gilbert Mozzo, Nov 21 2011
a(29) corresponds to a probable prime discovered by Paul Bourdelais, Apr 25 2019
a(30) corresponds to a probable prime discovered by Paul Bourdelais, Aug 12 2019
a(31) corresponds to a probable prime discovered by Paul Bourdelais, Jun 18 2020

A247093 Triangle read by rows: T(m,n) = smallest odd prime p such that (m^p-n^p)/(m-n) is prime (0

Original entry on oeis.org

3, 3, 3, 0, 0, 3, 3, 5, 13, 3, 3, 0, 0, 0, 5, 5, 3, 3, 5, 3, 3, 3, 0, 3, 0, 19, 0, 7, 0, 3, 0, 0, 3, 0, 3, 7, 19, 0, 3, 0, 0, 0, 31, 0, 3, 17, 5, 3, 3, 5, 3, 5, 7, 5, 3, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 5, 3, 7, 5, 5, 3, 7, 3, 3, 251, 3, 17, 3, 0, 5, 0, 151, 0, 0, 0, 59, 0, 5, 0, 3, 3, 5, 0, 1097, 0, 0, 3, 3, 0, 0, 7, 0, 17, 3

Offset: 1

Eric Chen, Nov 18 2014

T(m,n) is 0 if and only if m and n are not coprime or A052409(m) and A052409(n) are not coprime. (The latter has some exceptions, like T(8,1) = 3. In fact, if p is a prime and does not equal to A052410(gcd(A052409(m),A052409(n))), then (m^p-n^p)/(m-n) is composite, so if it is not 0, then it is A052410(gcd(A052409(m),A052409(n))).) - Eric Chen, Nov 26 2014
a(i) = T(m,n) corresponds only to probable primes for (m,n) = {(15,4), (18,1), (19,18), (31,6), (37,22), (37,25), ...} (i={95, 137, 171, 441, 652, 655, ...}). With the exception of these six (m,n), all corresponding primes up to a(663) are definite primes. - Eric Chen, Nov 26 2014
a(n) is currently known up to n = 663, a(664) = T(37, 34) > 10000. - Eric Chen, Jun 01 2015
For n up to 1000, a(n) is currently unknown only for n = 664, 760, and 868. - Eric Chen, Jun 01 2015

			Read by rows:
m\n        1   2   3   4   5   6   7   8   9   10  11
2          3
3          3   3
4          0   0   3
5          3   5   13  3
6          3   0   0   0   5
7          5   3   3   5   3   3
8          3   0   3   0   19  0   7
9          0   3   0   0   3   0   3   7
10         19  0   3   0   0   0   31  0   3
11         17  5   3   3   5   3   5   7   5   3
12         3   0   0   0   3   0   3   0   0   0   3
etc.

Eric Chen, Table of n, a(n) for n = 1..663
Eric Chen, Table of n, a(n) for n = 1..1000 status

Cf. A128164 (n,1), A125713 (n+1,n), A125954 (2n+1,2), A122478 (2n+1,2n-1).

Cf. A000043 (2,1), A028491 (3,1), A057468 (3,2), A059801 (4,3), A004061 (5,1), A082182 (5,2), A121877 (5,3), A059802 (5,4), A004062 (6,1), A062572 (6,5), A004063 (7,1), A215487 (7,2), A128024 (7,3), A213073 (7,4), A128344 (7,5), A062573 (7,6), A128025 (8,3), A128345 (8,5), A062574 (8,7), A173718 (9,2), A128346 (9,5), A059803 (9,8), A004023 (10,1), A128026 (10,3), A062576 (10,9), A005808 (11,1), A210506 (11,2), A128027 (11,3), A216181 (11,4), A128347 (11,5), A062577 (11,10), A004064 (12,1), A128348 (12,5), A062578 (12,11).

t1[n_] := Floor[3/2 + Sqrt[2*n]]
m[n_] := Floor[(-1 + Sqrt[8*n-7])/2]
t2[n_] := n-m[n]*(m[n]+1)/2
b[n_] := GCD @@ Last /@ FactorInteger[n]
is[m_, n_] := GCD[m, n] == 1 && GCD[b[m], b[n]] == 1
Do[k=2, If[is[t1[n], t2[n]], While[ !PrimeQ[t1[n]^Prime[k] - t2[n]^Prime[k]], k++]; Print[Prime[k]], Print[0]], {n, 1, 663}] (* Eric Chen, Jun 01 2015 *)

a052409(n) = my(k=ispower(n)); if(k, k, n>1);
a(m, n) = {if (gcd(m,n) != 1, return (0)); if (gcd(a052409(m), a052409(n)) != 1, return (0)); forprime(p=3,, if (isprime((m^p-n^p)/(m-n)), return (p)););}
tabl(nn) = {for (m=2, nn, for(n=1, m-1, print1(a(m,n), ", ");); print(););} \\ Michel Marcus, Nov 19 2014

t1(n)=floor(3/2+sqrt(2*n))
t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2)
b(n)=my(k=ispower(n)); if(k, k, n>1)
a(n)=if(gcd(t1(n),t2(n)) !=1 || gcd(b(t1(n)), b(t2(n))) !=1, 0, forprime(p=3,2^24,if(ispseudoprime((t1(n)^p-t2(n)^p)/(t1(n)-t2(n))), return(p)))) \\ Eric Chen, Jun 01 2015

A294722 Numbers k such that (44^k - 1)/43 is prime.

Original entry on oeis.org

5, 31, 167, 100511

Offset: 1

Paul Bourdelais, Nov 07 2017

The number corresponding to a(4) is a probable prime.

These are the indices of base-44 repunit primes, i.e., numbers k such that A002275(k) interpreted as a base-44 number and converted to decimal is prime. - Felix Fröhlich, Nov 08 2017

P. Bourdelais, A generalized repunit conjecture

Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765.

ParallelMap[ If[ PrimeQ[(44^# - 1)/43], #, Nothing] &, Prime@Range @ 10000] (* Robert G. Wilson v, Nov 25 2017 *)

is(n) = ispseudoprime((44^n-1)/43) \\ Felix Fröhlich, Nov 08 2017

ABC2 (44^$a-1)/43 // -f{2*$a}
a: primes from 2 to 1000000

A366582 Numbers k such that 6^k + 1 is a semiprime.

Original entry on oeis.org

3, 8, 11, 12, 31, 43, 47, 59, 62, 107, 382, 514, 734, 811

Offset: 1

Sean A. Irvine, Oct 13 2023

			11 is in this sequence because 6^11+1 = 7*51828151 is a semiprime.

Cf. A062394, A001358, A004062, A092559.

A366648 Numbers k such that 4^k + 1 is a semiprime.

Original entry on oeis.org

3, 6, 10, 14, 16, 20, 32, 46, 52, 64, 74, 128, 178, 298, 346, 502, 614, 634, 1912, 60394, 92116

Offset: 1

Sean A. Irvine, Oct 16 2023

			14 is in this sequence because 4^14+1 = 17*15790321 is a semiprime.

Cf. A062394, A001358, A004062, A092559.

The even terms of A092559 divided by 2. - Max Alekseyev, Jan 04 2024

a(19)-a(21) from Max Alekseyev, Jan 04 2024

cp's OEIS Frontend

A185073 Numbers n such that (34^n - 1)/33 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A198725 Primes of the form (6^n-11)/5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A199165 Numbers n such that (6^n-11)/5 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A247093 Triangle read by rows: T(m,n) = smallest odd prime p such that (m^p-n^p)/(m-n) is prime (0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A294722 Numbers k such that (44^k - 1)/43 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PFGW

A366582 Numbers k such that 6^k + 1 is a semiprime.

Views

Author

Keywords

Examples

Crossrefs

A366648 Numbers k such that 4^k + 1 is a semiprime.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions