A004064 -id:A004064 - 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

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

A273814 Numbers k such that (12^k - 7^k)/5 is prime.

Original entry on oeis.org

2, 3, 7, 13, 47, 89, 139, 523, 1051

Offset: 1

Tim Johannes Ohrtmann, May 31 2016

All terms are prime.
The corresponding primes are 19, 277, 7001653, 21379263273733, 105329145253605843602014309589572596276217801876213, ...
a(10) > 50000. - Michael S. Branicky, Nov 12 2024

Cf. A004064, A062578, A128348.

A273814:=n->`if`(isprime((12^n - 7^n)/5), n, NULL): seq(A273814(n), n=1..10^3); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[1, 10000], PrimeQ[(12^# - 7^#)/5] &]

for(n=1, 10000, if(isprime((12^n - 7^n)/5), print1(n, ", ")))

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

A250210 Irregular triangle read by rows in which row n lists the prime factors of the duodecimal repunit ((12^n-1)/11). (Written in base 10).

Original entry on oeis.org

13, 157, 5, 13, 29, 22621, 7, 13, 19, 157, 659, 4943, 5, 13, 29, 89, 233, 37, 157, 80749, 13, 19141, 22621, 11, 23, 266981089, 5, 7, 13, 19, 29, 157, 20593, 477517, 20369233, 13, 211, 659, 4943, 13063, 61, 157, 661, 9781, 22621, 5, 13, 17, 29, 89, 97, 233, 260753

Offset: 2

Eric Chen, Dec 29 2014

			Triangle begins:
[13]
[157]
[5, 13, 29]
[22621]
[7, 13, 19, 157]
[659, 4943]
[5, 13, 29, 89, 233]
[37, 157, 80749]
[13, 19141, 22621]
...

Cf. A102380, A004064, A250288.

tabf(nn) = for (n=1, nn, print(factor((12^n-1)/11)[,1]~);); \\ Michel Marcus, Dec 29 2014

cp's OEIS Frontend

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

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

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A273814 Numbers k such that (12^k - 7^k)/5 is prime.

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

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A250210 Irregular triangle read by rows in which row n lists the prime factors of the duodecimal repunit ((12^n-1)/11). (Written in base 10).

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A366702 Numbers k such that 12^k + 1 is a semiprime.

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