A062585 -id:A062585 - cp's OEIS frontend

A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

3, 3, 3, 3, 5, 3, 7, 7, 3, 3, 3, 17, 3, 3, 43, 5, 3, 1607, 5, 19, 127, 229, 3, 3, 3, 13, 3, 3, 149, 3, 5, 3, 23, 3, 5, 83, 3, 3, 37, 7, 3, 3, 37, 5, 3, 5, 58543, 3, 3, 7, 29, 3, 479, 5, 3, 19, 5, 3, 4663, 54517, 17, 3, 3, 5, 7, 3, 3, 17, 11, 47, 61, 19, 23, 3, 5, 19, 7, 5, 7, 3, 3

Offset: 1

Alexander Adamchuk, Dec 01 2006, Feb 15 2007

Corresponding smallest primes of the form (n+1)^p - n^p, where p = a(n) is an odd prime, are listed in A121091(n+1) = {7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, ...}. a(n) = A058013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1. a(97) = 7. a(99)..a(112) = {5, 43, 5, 13, 7, 5, 3, 6529, 59, 3, 5, 5, 113, 5}. a(114) = 139. a(117)..a(129) = {7, 13, 3, 5, 5, 7, 3, 5167, 3, 41, 59, 3, 3}. a(131) = 101. a(n) is currently unknown for n = {113, 115, 116, 130, 132, ...}.
a(96) = 1307, a(98) = 709.
a(137) is probably 196873 from a prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - Robert Price, Feb 17 2012
a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - Ray Chandler, Feb 26 2017

Robert Price and Ray Chandler, Table of n, a(n) for n = 1..138 (first 136 terms from Robert Price)
Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).
Cf. A065913 (smallest prime of form (n+1)^k - n^k).
Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).
Cf. A047845, A014076.
Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).
Cf. A000043, A057468, A059801, A059802, A062572-A062666.

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

A121091 Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime.

Original entry on oeis.org

7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, 5613125740675652943160572913465695837595324940170321, 371281, 919

Offset: 2

Alexander Adamchuk, Aug 11 2006, revised Dec 01 2006, Feb 15 2007

nonn

a(19) = 19^1607 - 18^1607, which is too large to include. It has 2055 decimal digits. See A062585(1) = 1607.

a(20)-a(21) = {723901, 8005616640331026125580781}. a(n) is currently known for all n up to n = 96. Corresponding smallest odd primes p such that (n+1)^p - n^p is prime are listed in A125713(n) = {3,3,3,3,5,3,7,7,3,3,3,17,3,3,43,5,3,10957,5,19,127,229,3,3,3,13,3,3,149,3,5,3,23,3,5,83,3,3,37,7,3,3,37,5,3,5,58543,...}. a(n+1) = A065013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1.

Cf. A121620, A000043, A058765, A121616, A121618.
Cf. A125713 = Smallest odd prime p such that (n+1)^p - n^p is prime. Cf. A065913 = Smallest prime of form (n+1)^k - n^k. Cf. A058013 = Smallest prime p such that (n+1)^p - n^p is prime. Cf. A047845, A014076.
Cf. A062585 = numbers n such that k^n - (k-1)^n is prime, where k is 19. Cf. A000043, A057468, A059801, A059802, A062572-A062666.

a(n) = n^A125713(n) - (n-1)^A125713(n).

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

cp's OEIS Frontend

A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

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

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A121091 Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime.

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

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