A004061 -id:A004061 - cp's OEIS frontend

A245237 Numbers k such that (48^k - 1)/47 is prime.

19, 269, 349, 383, 1303, 15031, 200443, 343901

Offset: 1

Robert Price, Jul 14 2014

a(7) > 10^5.

All terms are prime.

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765, A242797, A243279.

A245237:=n->`if`(isprime((48^n - 1)/47), n, NULL); seq(A245237(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(48^# - 1)/47] &]

is(n)=ispseudoprime((48^n-1)/47) \\ Charles R Greathouse IV, Jun 06 2017

a(7) corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

a(8) from Paul Bourdelais, Mar 03 2025

A245442 Numbers n such that (50^n - 1)/49 is prime.

Original entry on oeis.org

3, 5, 127, 139, 347, 661, 2203, 6521, 210319

Offset: 1

Robert Price, Jul 22 2014

a(9) > 10^5.

All terms are prime.

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765, A242797, A243279, A245237.

A245442:=n->`if`(isprime((50^n - 1)/49), n, NULL); seq(A245442(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(50^# - 1)/49] &]

is(n)=ispseudoprime((50^n-1)/49) \\ Charles R Greathouse IV, Jun 13 2017

a(9)=210319 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

A275938 Numbers m such that d(m) is prime while sigma(m) is not prime (where d(m) = A000005(m) and sigma(m) = A000203(m)).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Altug Alkan, Aug 12 2016

From Robert Israel, Aug 12 2016: (Start)
d(m) is prime iff m = p^k where p is prime and k+1 is prime.
For such m, sigma(m) = 1 + p + ... + p^k = (p*m-1)/(p-1).
The sequence contains 2^(q-1) for q in A054723,
3^(q-1) for q prime but not in A028491,
5^(q-1) for q prime but not in A004061,
7^(q-1) for q prime but not in A004063, etc.
In particular, it contains all odd primes. (End)

			49 is a term because A000005(49) = 3 is prime while sigma(49) = 57 is not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A004061, A004063, A009087, A023194, A028491.

Cf. A000040, A286095.

N:= 1000: # to get all terms <= N
P:= select(isprime, {2,seq(p,p=3..N,2)}):
fp:= proc(p) local q,res;
  q:= 2;
  res:= NULL;
  while p^(q-1) <= N do
     if not isprime((p^q-1)/(p-1)) then res:= res, p^(q-1) fi;
     q:= nextprime(q);
  od;
  res;
end proc:
sort(convert(map(fp, P),list)); # Robert Israel, Aug 12 2016

lista(nn) = for(n=1, nn, if(isprime(numdiv(n)) && !isprime(sigma(n)), print1(n, ", ")));

UNION of A000040 and A286095 (except for the term 2). - Bill McEachen, Jul 16 2024

A088554 Primes of the form (5^k-1)/4 or (5^k+3)/4.

Original entry on oeis.org

2, 7, 31, 157, 3907, 19531, 2441407, 12207031, 61035157, 305175781, 9313225746154785157, 177635683940025046467781066894531

Offset: 1

Cino Hilliard, Nov 17 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..19

Cf. A004061.

With[{kmax = 100}, Select[Flatten[Table[(5^k + {-1, 3})/4, {k, 0, kmax}]], PrimeQ]] (* Amiram Eldar, Jul 05 2024 *)

Edited by Don Reble, May 03 2006

A181987 Numbers n such that (39^n - 1)/38 is prime.

Original entry on oeis.org

349, 631, 4493, 16633, 36341

Offset: 1

Robert Price, Apr 04 2012

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

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

Select[Prime[Range[100000]], PrimeQ[(39^#-1)/38]&]

is(n)=ispseudoprime((39^n-1)/38) \\ Charles R Greathouse IV, Jun 13 2017

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

Original entry on oeis.org

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

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

A140509 Numbers k such that 5^k-1 contains a divisor which is an overpseudoprime to base 5.

Original entry on oeis.org

5, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Vladimir Shevelev, Jun 30 2008

nonn

An odd prime p is in the sequence iff p is not in A004061.

Cf. A141390, A020231, A141232, A141350, A003463, A004061.

isokd(n) = (n>5) && !isprime(n) && (gcd(n,5)==1) && (znorder(Mod(5,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(5, d))) - 1) + 1 == n); \\ A141390
isok(n) = {fordiv (5^n-1, dd, if (isokd(dd), return (1));); return (0);} \\ Michel Marcus, Oct 25 2018

Corrected and more terms from Michel Marcus, Oct 25 2018

cp's OEIS Frontend

A245237 Numbers k such that (48^k - 1)/47 is prime.

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A245442 Numbers n such that (50^n - 1)/49 is prime.

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A275938 Numbers m such that d(m) is prime while sigma(m) is not prime (where d(m) = A000005(m) and sigma(m) = A000203(m)).

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A088554 Primes of the form (5^k-1)/4 or (5^k+3)/4.

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A181987 Numbers n such that (39^n - 1)/38 is prime.

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

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