A059801 -id:A059801 - cp's OEIS frontend

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

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

A286348 Numbers n such that 4^n + (-3)^n is prime.

Original entry on oeis.org

0, 3, 4, 7, 16, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233

Offset: 1

Juri-Stepan Gerasimov, May 07 2017

Numbers n such that (1 + k)^n + (-k)^n is prime:
0 (k = 0);
A285929 (k = 1);
A283653 (k = 2);
this sequence (k = 3);
0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);
0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ...(k = 5);
0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);
0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);
0, 4, 7, 29, 31, 32, 67, ... (k = 8);
0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);
0, 3, 5, 19, 32, ... (k = 10);
0, 3, 7, 89, 101, ... (k = 11);
0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);
0, 3, 4, 11, 83, ... (k = 13);
0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);
0, 43, ... (k = 15);
0, 4, 5, 7, 79, ... (k = 16);
0, 2, 3, 8, 13, 71, ... (k = 17);
0, 1607, ... (k = 18);
...
Primes of the form (1 + n)^(2^n) + n: 5, 83, 65539, 7958661109946400884391941, ...
Numbers m such that (1 + k)^m + (-k)^m is not odd prime for k =< m: 0, 1, 15, 18, 53, 59, 106, 114, 124, 132, 133, 143, 177, 214, 232, 234, 240, 256, ...
Conjecture: if (1 + y)^x + (-y)^x is a prime number then x is zero, or an even power of two, or an odd prime number.

			3 is in this sequence because 4^3 + (-3)^3 = 37 is prime.
4 is in this sequence because 4^4 + (-3)^4 = 337 is prime.

Cf. A059801, A081505, A283653, A285929.

[n: n in [0..250] | IsPrime(4^n+(-3)^n)];

Select[Range[0, 3000], PrimeQ[4^# + (-3)^#] &] (* Michael De Vlieger, May 09 2017 *)

is(n)=ispseudoprime(4^n+(-3)^n) \\ Charles R Greathouse IV, Jun 13 2017

A062580 Numbers k such that 14^k - 13^k is prime.

Original entry on oeis.org

3, 11, 83, 461, 659, 1129, 3797, 83869

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms greater than 1000 may correspond to (unproven) strong pseudoprimes.

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(14^n-13^n) \\ Charles R Greathouse IV, Jun 13 2017

83869 found from Jean-Louis Charton, Sep 02 2009

Edited by M. F. Hasler, Sep 16 2013

A062594 Numbers k such that 28^k - 27^k is prime.

Original entry on oeis.org

3, 5, 19, 31, 257, 773

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 are often only strong pseudoprimes.

a(7) > 10^5. - Robert Price, Sep 04 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(28^n-27^n) \\ Charles R Greathouse IV, Jun 13 2017

A062595 Numbers k such that 29^k - 28^k is prime.

Original entry on oeis.org

3, 7219, 34871

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 are often only strong pseudoprimes.

a(4) > 10^5. - Robert Price, Sep 22 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(29^n-28^n) \\ Charles R Greathouse IV, Jun 13 2017

a(3) from Robert Price, Sep 22 2012

A062596 Numbers k such that 30^k - 29^k is prime.

Original entry on oeis.org

2, 149, 283, 853, 1741, 4831, 8867

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 are often only strong pseudoprimes.

a(8) > 10^5. - Robert Price, Oct 23 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(30^n-29^n) \\ Charles R Greathouse IV, Jun 13 2017

A062597 Numbers k such that 31^k - 30^k is prime.

Original entry on oeis.org

2, 3, 5, 211

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 are often only strong pseudoprimes.

a(5) > 10^5 - Robert Price, Nov 14 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(31^n-30^n) \\ Charles R Greathouse IV, Jun 13 2017

A062598 Numbers k such that 32^k - 31^k is prime.

Original entry on oeis.org

5, 1427, 2357, 24499

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 often correspond only to strong pseudoprimes.

a(5) > 10^5. - Robert Price, Oct 03 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(32^n-31^n) \\ Charles R Greathouse IV, Jun 13 2017

a(4) from Robert Price, Oct 03 2012

cp's OEIS Frontend

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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A286348 Numbers n such that 4^n + (-3)^n is prime.

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A062580 Numbers k such that 14^k - 13^k is prime.

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A062594 Numbers k such that 28^k - 27^k is prime.

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A062595 Numbers k such that 29^k - 28^k is prime.

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A062596 Numbers k such that 30^k - 29^k is prime.

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A062597 Numbers k such that 31^k - 30^k is prime.

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