A128026 -id:A128026 - cp's OEIS frontend

A173718 Numbers n such that (9^n - 2^n)/7 is prime.

2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, 108553, 200807

Offset: 1

Robert Price, Dec 22 2012

All terms are prime.

a(14) > 10^6.

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Select[ Prime[ Range[1, 100000] ], PrimeQ[ (9^# - 2^#)/7 ]& ]

is(n)=ispseudoprime((9^n-2^n)/7) \\ Charles R Greathouse IV, Jun 06 2017

a(12)-a(13) from Jon Grantham, Jul 29 2023

A128033 Least number k>0 such that ((n+3)^k - 3^k)/n is prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 2, 13, 0, 3, 2, 0, 2, 3, 0, 7, 2, 0, 2, 3, 0, 73, 2, 0, 5, 3, 0, 3, 2, 0, 2, 3, 0, 3, 3, 0, 2, 5, 0, 3, 2, 0, 2, 401, 0, 3, 2, 0, 5, 5, 0, 3, 2, 0

Offset: 0

Alexander Adamchuk, Feb 11 2007

All positive terms are primes.
a(50)-a(67) = {7, 0, 79, 2, 0, 2, 109, 0, 5, 5, 0, 2, 5, 0, 131, 2, 0, 2}. a(69)-a(121) = {0, 3, 19, 0, 2, 5, 0, 11, 2, 0, 13, 7, 0, 3, 2, 0, 3, 11, 0, 3, 19, 0, 2, 3, 0, 11, 2, 0, 2, 3, 0, 17, 2, 0, 2, 3, 0, 5, 2, 0, 3, 31, 0, 17, 5, 0, 47, 31, 0, 3, 3, 0, 2}.
a(49) > 10000. - Jinyuan Wang, Nov 28 2020

Cf. A128049 (least number k>0 such that abs((3^k - (3-n)^k)/n) is prime), A028491, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

a(n) = my(p=2); if(n%3, while(!ispseudoprime(((n+3)^p-3^p)/n), p=nextprime(p+1)); p, 0); \\ Jinyuan Wang, Nov 28 2020

a(3*n) = 0.

A215487 Numbers k such that (7^k - 2^k)/5 is prime.

Original entry on oeis.org

3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, 341063, 508867, 720497, 846913

Offset: 1

Robert Price, Aug 12 2012

All terms are prime.

a(14) > 10^6.

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Select[ Prime[ Range[1, 300] ], PrimeQ[ (7^# - 2^#)/5 ]& ]

is(n)=ispseudoprime((7^n-2^n)/5) \\ Charles R Greathouse IV, Jun 13 2017

a(10)-a(13) from Jon Grantham, Jul 29 2023

A224691 Numbers n such that (13^n - 4^n)/9 is prime.

Original entry on oeis.org

2, 5, 19, 109, 157, 8521, 26017, 26177

Offset: 1

Robert Price, Apr 15 2013

All terms are prime.

a(9) > 10^5.

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032, A210506, A128347.

Select[Prime[Range[1, 100000]], PrimeQ[(13^# - 4^#)/9]&]

is(n)=ispseudoprime((13^n-4^n)/9) \\ Charles R Greathouse IV, Jun 13 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

A138931 Primes of the form (10^n-3^n)/7.

Original entry on oeis.org

13, 139, 14251, 1428571428571428571364245156301286091

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 16 2008

nonn

The next term is too large to include here.

Cf. A128026.

a={};Do[p=(10^n-3^n)/7;If[PrimeQ[p],AppendTo[a,p]],{n,1,25^2}];a
Select[Table[(10^n-3^n)/7,{n,50}],PrimeQ] (* Harvey P. Dale, Dec 18 2018 *)

A230139 Numbers n such that (17^n - 4^n)/13 is prime.

Original entry on oeis.org

3, 5, 7, 11, 31, 101, 887, 4861

Offset: 1

Robert Price, Oct 10 2013

All terms are prime.

a(9) > 10^5.

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032, A210506, A128347, A128352, A225807.

Select[Prime[Range[1, 100000]], PrimeQ[(17^# - 4^#)/13]&]

is(n)=ispseudoprime((17^n-4^n)/13) \\ Charles R Greathouse IV, Jun 13 2017

A241921 Numbers k such that (15^k - 4^k)/11 is prime.

Original entry on oeis.org

2, 1097, 2243, 2857, 4357, 6803, 20747, 24571

Offset: 1

Robert Price, May 01 2014

All terms are prime.
a(9) > 10^5.
a(1) is trivially prime, the remainder are probable primes.

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032, A210506, A128347, A225955, A062581.

Select[Prime[Range[1, 100000]], PrimeQ[(15^# - 4^#)/11]&]

is(n)=ispseudoprime((15^n-4^n)/11) \\ Charles R Greathouse IV, Jun 13 2017

A273403 Numbers k such that (10^k - 7^k)/3 is prime.

Original entry on oeis.org

2, 31, 103, 617, 10253, 10691, 35353, 129587, 556441

Offset: 1

Tim Johannes Ohrtmann, May 21 2016

All terms are prime.

The corresponding primes: 17, 3333280741539321718064461652419, ...

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A004023, A128026, A062576.

Select[Range[1, 10000], PrimeQ[(10^# - 7^#)/3] &]

for(n=1, 10000, if(isprime((10^n - 7^n)/3), print1(n, ", ")))

a(7)-a(9) from Jon Grantham, Jul 29 2023

A375620 Numbers k such that (20^k - 3^k)/17 is prime.

Original entry on oeis.org

2, 43, 1723, 2971, 3257, 12263, 38933

Offset: 1

William Dean, Aug 21 2024

All terms must be prime.

a(8) > 10^5.

			a(1) = 2 corresponds to the prime number 23.

OEIS Wiki, Primes of the form (a^n+b^n)/(a+b) and (a^n-b^n)/(a-b)

Cf. A028491, A057468, A128032, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

PARI
```
is(k) = ispseudoprime((20^k - 3^k)/17)
```

cp's OEIS Frontend

A173718 Numbers n such that (9^n - 2^n)/7 is prime.

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A128033 Least number k>0 such that ((n+3)^k - 3^k)/n is prime, or 0 if no such prime exists.

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A215487 Numbers k such that (7^k - 2^k)/5 is prime.

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A224691 Numbers n such that (13^n - 4^n)/9 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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A138931 Primes of the form (10^n-3^n)/7.

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A230139 Numbers n such that (17^n - 4^n)/13 is prime.

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A241921 Numbers k such that (15^k - 4^k)/11 is prime.

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A273403 Numbers k such that (10^k - 7^k)/3 is prime.