A121091 -id:A121091 - 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.

A124155 19^p - 18^p, where p = Prime[n].

Original entry on oeis.org

37, 1027, 586531, 281651707, 52221848818987, 21230018596585891, 3294475298046105653971, 1270184310304975912766347, 183481914331285799334907290427, 9601090905261258491400850200348915811

Offset: 1

Alexander Adamchuk, Dec 01 2006

nonn

The first prime in a(n) is a(1) = 37 = 19^2 - 18^2. The second prime in a(n) is a(1331) = 19^10957 - 18^10957. It has 14012 decimal digits. Note that 1331 = 11^3, Prime[11^3] = 10957. Last digit of a(n) is 1 or 7. It appears that 3^2 divides a(n) - 1. Also it appears that 7^3 divides all a(n) - 1 for n>2. a(n) - 1 is divisible by 7^m, where m(n) = {0,0,3,7,3,6,3,6,3,3,6,6,3,...}. 7^6 divides a(n) - 1 for n = {4,6,8,11,12,14,18,19,21,22,25,27,29,31,34,36,37,38,42,44,46,47,48,50,53,58,59,61,63,65,67,68,70,73,74,75,78,80,82,84,85,88,90,93,95,99,100,...}. 7^7 divides a(n) - 1 for n = {4,14,31,47,68,75,82,90,101,115,122,134,153,163,169,177,183,213,226,233,251,269,295,...}. 7^8 divides a(n) - 1 for n = {153,233,383,493,531,669,775,839,871,907,937,...}.

Cf. A121091 = Smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime. Cf. A125713 = Smallest odd prime p such that (n+1)^p - n^p is prime.

Mathematica
```
Table[19^Prime[n]-18^Prime[n],{n,1,15}]
```

a(n) = 19^Prime[n] - 18^Prime[n].

A321616 Primes p = k^2 + (k-1)^2 such that k^p - (k-1)^p is prime.

Original entry on oeis.org

5, 61, 113, 1741

Offset: 1

Thomas Ordowski, Nov 15 2018

Conjecture: generally, these are primes p = a^2 + b^2 with a > b > 0 such that (a^p - b^p)/(a-b) is prime, so must be a-b = 1. It seems that there are no primes (a^q + b^q)/(a+b) for primes q = a^2 + b^2 > 5. Especially, there are probably no primes q = m^2 + 1 > 5 such that (m^q - 1)/(m-1) is prime or (m^q + 1)/(m+1) is prime. How to prove it?
No more terms up to the prime 19801 = 100^2 + 99^2. - Amiram Eldar, Nov 15 2018
a(5) > 109045. - J.W.L. (Jan) Eerland, Dec 11 2022
a(5) > 209305. - Michael S. Branicky, Aug 21 2024

			The prime 5 = 2^2 + 1^2 and 2^5 - 1^5 = 31 is prime.
We have 61 = 6^2 + 5^2, 113 = 8^2 + 7^2, 1741 = 30^2 + 29^2.

Cf. A002144, A121091.

Subsequence of A027862.

f[k_]:=k^2 + (k-1)^2 ; seqQ[k_]:=Module[{p=f[k]}, PrimeQ[p] && PrimeQ[k^p - (k-1)^p ]]; f[Select[Range[30], seqQ]] (* Amiram Eldar, Nov 15 2018 *)
pQ[k_]:=Module[{c=k^2+(k-1)^2},If[AllTrue[{c,k^c-(k-1)^c},PrimeQ],c,Nothing]]; Array[pQ,30] (* Harvey P. Dale, Aug 27 2023 *)

lista(nn) = {for (k=1, nn, if (isprime(p=k^2 + (k-1)^2) && isprime(k^p - (k-1)^p), print1(p, ", ")););} \\ Michel Marcus, Nov 18 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

A124155 19^p - 18^p, where p = Prime[n].

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A321616 Primes p = k^2 + (k-1)^2 such that k^p - (k-1)^p is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI