A121620 -id:A121620 - cp's OEIS frontend

A121616 Primes of form (k+1)^5 - k^5 = A022521(k).

31, 211, 4651, 61051, 371281, 723901, 1803001, 2861461, 4329151, 4925281, 7086451, 7944301, 14835031, 19611901, 23382031, 44119351, 54664711, 86548801, 97792531, 162478501, 189882031, 267217051, 293109961, 306740281, 490099501

Offset: 1

Alexander Adamchuk, Aug 10 2006

nonn

Might be called "Pentan primes" (in analogy with Cuban primes, of the form (n+1)^3-n^3), or "Nexus primes of order 5" (cf. link below).
Indices k such that Nexus number of order 5 (or A022521(k-1) = k^5 - (k-1)^5) is prime are listed in A121617 = {2, 3, 6, 11, 17, 20, 25, 28, 31, 32, 35, 36, 42, 45, 47, 55, 58, 65, 67, 76, 79, 86, 88, 89, 100,...}.
The last digit is always 1 because 5 is the Pythagorean prime A002144(1). a(1) = 31 is the Mersenne prime A000668(3).

Zak Seidov, Table of n, a(n) for n = 1..2000.

Cf. A022521, A000040, A000043, A002144, A000668, A002407, A121617, A121618, A121619, A121620.

[a: n in [0..110] | IsPrime(a) where a is (n+1)^5-n^5]; // Vincenzo Librandi, Jan 20 2020

Select[Table[n^5 - (n-1)^5, {n,1,200}],PrimeQ]
Select[Differences[Range[100]^5],PrimeQ] (* Harvey P. Dale, Nov 03 2021 *)

A121618 Nexus primes of order 7 or primes of form n^7 - (n-1)^7 = A022523(n-1).

Original entry on oeis.org

127, 14197, 543607, 1273609, 2685817, 5217031, 16344637, 141903217, 1928294551, 8258704609, 14024867221, 22815424087, 30914273881, 91154730577, 116160677851, 183510024391, 280667317267, 552932810431, 1517045588059

Offset: 1

Alexander Adamchuk, Aug 10 2006

Indices n such that Nexus number of 7 order (or A022523(n-1) = n^7 - (n-1)^7) is prime are listed in A121619(n) = {2,4,7,8,9,10,12,17,26, 33,36,39,41,49,51,55,59,66,78,79,80,88,96,98,...}. a(1) = 127 is Mersenne prime A000668(4).

Vladimir Pletser, Table of n, a(n) for n = 1..1000

Cf. A022523, A000040, A000043, A000668, A002407, A121616, A121618, A121619, A121620.

Select[Table[n^7 - (n-1)^7, {n,1,200}],PrimeQ]
Select[Differences[Range[100]^7],PrimeQ] (* Harvey P. Dale, Mar 27 2025 *)

select(isprime,vector(1000,n,n^7-(n-1)^7)) \\ Charles R Greathouse IV, May 15 2013

A121619 Indices n such that Nexus numbers of order 7 (A022523(n-1) = n^7 - (n-1)^7) are primes.

Original entry on oeis.org

2, 4, 7, 8, 9, 10, 12, 17, 26, 33, 36, 39, 41, 49, 51, 55, 59, 66, 78, 79, 80, 88, 96, 98, 104, 113, 118, 120, 123, 135, 136, 142, 146, 156, 157, 160, 162, 173, 176, 194, 210, 219, 220, 221, 222, 224, 232, 234, 247, 281, 291, 297, 298, 305

Offset: 1

Alexander Adamchuk, Aug 10 2006

nonn

Corresponding Nexus primes of order 7 (or primes of form A022523(n-1) = n^7 - (n-1)^7) are listed in A121618[n] = {127, 14197, 543607, 1273609, 2685817, 5217031, 16344637, 141903217,...}.

Vladimir Pletser, Table of n, a(n) for n = 1..1000

Cf. A022523, A121619, A121618, A121616, A121620.

t = {}; Do[np7 = n^7 - (n - 1)^7; If[PrimeQ[np7], AppendTo[t, n]], {n, 305}]; t (* Carl R. White, Feb 28 2008 *)
Flatten[Position[Partition[Range[400]^7,2,1],?(PrimeQ[#[[2]]- #[[1]]]&), {1}, Heads->False]]+1 (* or *) Select[Range[400],PrimeQ[#^7-(#-1)^7]&] (* _Harvey P. Dale, Mar 19 2017 *)

More terms from Carl R. White, Feb 28 2008

A121617 Numbers n such that A022521(n-1) = n^5 - (n-1)^5 is prime.

Original entry on oeis.org

2, 3, 6, 11, 17, 20, 25, 28, 31, 32, 35, 36, 42, 45, 47, 55, 58, 65, 67, 76, 79, 86, 88, 89, 100, 102, 105, 110, 111, 113, 121, 122, 145, 149, 166, 175, 179, 193, 198, 211, 218, 223, 226, 230, 240, 244, 245, 256, 262, 287, 292, 295, 297, 298, 300

Offset: 1

Alexander Adamchuk, Aug 10 2006

nonn

The elements of A022521 are sometimes called Nexus number of order 5, see there.
The terms should have 1 subtracted, since indices of primes in A022521 are 1, 2, 5, 10, 16, 19, 24, 27, 30, 31, 34, 35, 41, 44, 46, .... - M. F. Hasler, Jan 27 2013
Corresponding Nexus Primes of order 5 (or primes of form (n+1)^5 - n^5 = A022521(n)) are listed in A121616 = {31, 211, 4651, 61051, 371281, 723901, 1803001, 2861461, ...}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A022521, A121616, A121618, A121619, A121620.

select(t -> isprime(t^5-(t-1)^5), [$1..1000]); # Robert Israel, Jul 10 2018

Do[np5=n^5 - (n-1)^5; If[PrimeQ[np5],Print[n]],{n,1,100}]
Flatten[Position[Partition[Range[300]^5,2,1],?(PrimeQ[#[[2]]-#[[1]]]&), 1,Heads-> False]]+1 (* _Harvey P. Dale, May 30 2021 *)

A121617(n,print_all=0)={for(k=2,9e9, ispseudoprime(k^5-(k-1)^5) & !(print_all & print1(k",")) & !n-- & return(k))} \\ M. F. Hasler, Feb 03 2013

A222119 Number k yielding the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, 402, 3, 44, 10, 82, 20, 95, 4, 108, 349, 127, 303, 37, 3, 162

Offset: 1

Vladimir Pletser, Feb 07 2013

nonn

The smallest k generating a prime of the form (k+1)^p - k^p (A121620) for the prime A000040(n). For the primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... (A000043), k = 1 and Mersenne primes 2^p - 1 (A000668) are obtained. For p = 11, 23, 29, ..., the smallest primes of the form (k+1)^p - k^p are respectively 313968931 (for k = 5), 777809294098524691 (for k = 5 also), 68629840493971 (for k = 2), ..., so a(5) = 5, a(9) = 5, a(10) = 2, ...

Jinyuan Wang, Table of n, a(n) for n = 1..175

Cf. A103794, A222120 (number of digits in the primes).

A222119 := proc(n)
        p := ithprime(n) ;
        for k from 1 do
                if isprime((k+1)^p-k^p) then
                        return k;
                end if;
        end do:
end proc: # R. J. Mathar, Feb 10 2013

Table[p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; k, {n, 80}] (* T. D. Noe, Feb 12 2013 *)

f(p) = {my(k=1); while(ispseudoprime((k+1)^p-k^p)==0, k++); k; }
lista(nn) = forprime(p=2, nn, print1(f(p), ", ")); \\ Jinyuan Wang, Feb 03 2020

a(n) = A103794(n) - 1. - Ray Chandler, Feb 26 2017

More terms from Ray Chandler, Feb 27 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).

A222120 Number of digits in the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

Original entry on oeis.org

1, 1, 2, 3, 9, 4, 6, 6, 18, 14, 10, 60, 35, 31, 53, 26, 29, 19, 57, 90, 122, 72, 65, 27, 138, 49, 168, 33, 122, 103, 39, 119, 345, 126, 143, 250, 225, 182, 315, 204, 308, 371, 134, 227, 335, 489, 255, 156, 364, 312, 476, 613, 329, 460, 372, 522, 514, 590, 133

Offset: 1

Vladimir Pletser, Feb 07 2013

The smallest primes of the form (k+1)^p - k^p are in A121620. The values of k are in A222119. For the primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... (A000043), k = 1 and Mersenne primes 2^p - 1 (A000668) are obtained.

			a(5) = 9 because the 5th prime is 11, and the smallest prime of the form (k+1)^11 - k^11 is 6^11 - 5^11 = 313968931, which has 9 digits

Cf. A222119 (values of k).

Table[p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; Length[IntegerDigits[q]], {n, 60}] (* T. D. Noe, Feb 12 2013 *)

a222120(n) = {local(p,k); p=prime(n); while(!isprime((k+1)^p - k^p), k=k+1); ceil(log((k+1)^p - k^p)/log(10))} \\ Michael B. Porter, Feb 12 2013

cp's OEIS Frontend

A121616 Primes of form (k+1)^5 - k^5 = A022521(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A121618 Nexus primes of order 7 or primes of form n^7 - (n-1)^7 = A022523(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A121619 Indices n such that Nexus numbers of order 7 (A022523(n-1) = n^7 - (n-1)^7) are primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A121617 Numbers n such that A022521(n-1) = n^5 - (n-1)^5 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A222119 Number k yielding the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Formula

A222120 Number of digits in the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI