A059803 -id:A059803 - cp's OEIS frontend

A062576 Numbers k such that 10^k - 9^k is prime.

2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 are often only strong pseudoprimes.

All terms are prime. - Alexander Adamchuk, Apr 27 2008

			10^2 - 9^2 = 100 - 81 = 19, which is prime, hence 2 is in the sequence.
10^3 - 9^3 = 1000 - 729 = 271, which is prime, hence 3 is in the sequence.
10^4 - 9^4 = 10000 - 6561 = 3439 = 19 * 181, which is not prime, hence 4 is not in the sequence.

Henri & Renaud Lifchitz, PRP Records.

Cf. A000043, A057468, A059801, A059802, A059803 (9^n-8^n is prime), A062572-A062666.

Cf. A016189 = 10^n - 9^n, and A199819 (primes of this form).

Select[Range[1000], PrimeQ[10^# - 9^#] &] (* Alonso del Arte, Sep 06 2013 *)

is(n)=ispseudoprime(10^n-9^n) \\ Charles R Greathouse IV, Feb 20 2017

Three more terms 15787, 66949 and 282493 found by Jean-Louis Charton in 2004 and 2007

A062574 Numbers k such that 8^k - 7^k is prime or a strong pseudoprime.

Original entry on oeis.org

7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

All terms are prime. - Alexander Adamchuk, Apr 27 2008

Henri & Renaud Lifchitz, PRP Records.

Cf. A000043, A057468, A059801, A059802, A059803 (9^n-8^n is prime), A062572-A062666.

Cf. A016177 = 8^n - 7^n.

Select[Range[200], PrimeQ[8^# - 7^#] &] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2012 *)

is(n)=ispseudoprime(8^n-7^n) \\ Charles R Greathouse IV, Sep 04 2013

Two more terms 44029 and 76213 found by Ananda Tallur & Jean-Louis Charton in 2003.
Three more terms 83663, 173687 and 336419 found by Jean-Louis Charton in 2004 and 2008
New term 615997 found by Jean-Louis Charton corresponding to a probable prime with 556301 digits. Jean-Louis Charton, Sep 02 2009

A062583 Numbers k such that 17^k - 16^k is prime.

Original entry on oeis.org

5, 7, 79, 523, 571, 2837

Offset: 1

Mike Oakes, May 18 2001

Terms greater than 1000 may only be strong pseudoprimes. [Clarified by M. F. Hasler, Sep 16 2013]

No other terms less than 100000. - Robert Price, Mar 22 2012

Cf. A000043, A057468, A059801-A059803, A062572-A062582, A188051, A062585-A062666.

is(n)=ispseudoprime(17^n-16^n) \\ Charles R Greathouse IV, May 22 2017

A214655 Numbers n such that 25^n - 24^n is prime or a strong pseudoprime.

Original entry on oeis.org

3, 5, 29, 54799

Offset: 1

Robert Price, Jul 24 2012

All terms are prime.

No other terms less than 10^5. - Robert Price, Jul 24 2012

Cf. A000043, A057468, A059801, A059802, A059803, A062572-A062589, A214658, A062592-A062666.

Select[Range[10^5], PrimeQ[25^#-24^# ]&]

Edited by M. F. Hasler, Sep 21 2013

A214658 Numbers n such that 24^n - 23^n is prime or a strong pseudoprime.

Original entry on oeis.org

2, 3, 31, 40519, 51061

Offset: 1

Robert Price, Jul 24 2012

All terms are prime.

No other terms less than 10^5. - Robert Price, Jul 24 2012

Cf. A000043, A057468, A059801, A059802, A059803, A062572-A062589, A214655, A062592-A062666.

Select[Range[10^5], PrimeQ[24^#-23^# ]&]

A062609 Numbers k such that 43^k - 42^k is prime or a strong pseudoprime.

Original entry on oeis.org

3, 13, 43, 211

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

a(5) > 10^5 - Robert Price, Dec 24 2012

Cf. A000043, A057468, A059801-A059803, A062572-A062608, A215632, A062611-A062666.

forprime(n=1, 9999, ispseudoprime(43^n-42^n) && print1(n", ")) \\ - M. F. Hasler, Sep 21 2013

Edited by M. F. Hasler, Sep 21 2013

A062611 Numbers k such that 45^k - 44^k is prime or a strong pseudoprime.

Original entry on oeis.org

2, 5, 151, 223, 313, 1277, 8447

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

a(8) > 10^5. - Robert Price, Jan 02 2013

Cf. A000043, A057468, A059801-A059803, A062572-A062609, A215632, A062612-A062666.

forprime(n=1,9999,ispseudoprime(45^n-44^n)&print1(n",")) \\ - M. F. Hasler, Sep 21 2013

Edited by M. F. Hasler, Sep 21 2013

A188051 Numbers k such that 18^k - 17^k is prime, or a strong pseudoprime.

Original entry on oeis.org

3, 13, 71, 14533, 26641, 48179

Offset: 1

Jean-Louis Charton, Mar 19 2011

Terms < 10000 found by Mike Oakes.
Term 14533 found by Lelio R Paula in June 2008 corresponding to a probable prime with 18243 digits.
Terms 26641 and 48179 found by Jean-Louis Charton in December 2010 corresponding to probable primes with 33442 and 60478 digits.
a(7) > 10^5. - Robert Price, Nov 30 2012

Top probable primes of the form 18^p-17^p

Cf. A000043, A057468, A059801, A059802, A059803, A062572-A062583, A062585-A062666.

is(n)=ispseudoprime(18^n-17^n) \\ Charles R Greathouse IV, Jun 06 2017

Edited by M. F. Hasler, Sep 21 2013

A215632 Numbers n such that 44^n - 43^n is prime or PRP.

Original entry on oeis.org

37, 283, 62903

Offset: 1

Jean-Louis Charton, Aug 18 2012

No other term <= 68141.

a(4) > 10^5. - Robert Price, Sep 04 2012

PRP Top Records - Search by form: 44^n-43^n

Cf. A000043, A057468, A059801-A059803, A062572-A062609, A062611-A062666.

is(n)=ispseudoprime(44^n-43^n) \\ Charles R Greathouse IV, Jun 06 2017

Edited by M. F. Hasler, Sep 21 2013

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

cp's OEIS Frontend

A062576 Numbers k such that 10^k - 9^k is prime.

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A062574 Numbers k such that 8^k - 7^k is prime or a strong pseudoprime.

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A062583 Numbers k such that 17^k - 16^k is prime.

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A214655 Numbers n such that 25^n - 24^n is prime or a strong pseudoprime.

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A214658 Numbers n such that 24^n - 23^n is prime or a strong pseudoprime.

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A062609 Numbers k such that 43^k - 42^k is prime or a strong pseudoprime.

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A062611 Numbers k such that 45^k - 44^k is prime or a strong pseudoprime.

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A188051 Numbers k such that 18^k - 17^k is prime, or a strong pseudoprime.

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A215632 Numbers n such that 44^n - 43^n is prime or PRP.

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