A121618 -id:A121618 - 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 *)

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

A121620 Smallest prime of the form k^p - (k-1)^p, where p = prime(n).

Original entry on oeis.org

3, 7, 31, 127, 313968931, 8191, 131071, 524287, 777809294098524691, 68629840493971, 2147483647, 114867606414015793728780533209145917205659365404867510184121, 44487435359130133495783012898708551, 1136791005963704961126617632861

Offset: 1

Alexander Adamchuk, Aug 10 2006

nonn

All Mersenne primes of form 2^p-1 = {3, 7, 31, 127, 8191,...} belong to a(n). Mersenne prime A000668(n) = a(k) when prime(k) = A000043(n). Last digit is always 1 for Nexus numbers of form n^p - (n-1)^p with p = {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101,...} = A004144(n) Pythagorean primes: primes of form 4n+1.

Vladimir Pletser and T. D. Noe, Table of n, a(n) for n = 1..80 (first 46 terms from Vladimir Pletser)

Cf. A121616, A121617, A121618, A121619, A022521, A022523, A004144, A000043, A000668.

t = {}; n = 0; While[n++; p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; q < 10^100, AppendTo[t, q]]; t (* T. D. Noe, Feb 12 2013 *)
spf[p_]:=Module[{k=2},While[CompositeQ[k^p-(k-1)^p],k++];k^p-(k-1)^p]; Table[spf[p],{p,Prime[ Range[20]]}] (* Harvey P. Dale, Apr 01 2024 *)

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

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).

A181126 Difference of two positive 7th powers.

Original entry on oeis.org

0, 127, 2059, 2186, 14197, 16256, 16383, 61741, 75938, 77997, 78124, 201811, 263552, 277749, 279808, 279935, 543607, 745418, 807159, 821356, 823415, 823542, 1273609, 1817216, 2019027, 2080768, 2094965, 2097024, 2097151, 2685817

Offset: 1

T. D. Noe, Oct 06 2010

nonn

Because x^7-y^7 = (x-y)(x^6+x^5*y+x^4*y^2+x^3*y^3+x^2*y^4+x*y^5+y^6), the difference of two 7th powers is a prime number only if x=y+1, in which case all the primes are in A121618.

The number 67675234241018881 = 127^8 is the first of an infinite number of squares of the form (b^(7k)-1)^8 in this sequence. Are any other squares possible?

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers)

nn=10^12; p=7; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]
Join[{0},#[[2]]-#[[1]]&/@Subsets[Range[10]^7,{2}]//Union] (* Harvey P. Dale, Oct 23 2024 *)

A221977 Number of primes of the form (x+1)^7 - x^7 less than 10^n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 7, 8, 10, 14, 18, 25, 34, 46, 60, 89, 120, 165, 227, 298, 415, 590, 821, 1152, 1606, 2240, 3188, 4438, 6208, 8714, 12280, 17368, 24560, 34821, 49413, 70581, 100856, 143955, 205291, 293061, 419256, 600213, 858870, 1230523, 1764914, 2532078

Offset: 1

Vladimir Pletser, Feb 02 2013

Number of primes less than 10^n and equal to the difference of two consecutive seventh powers (x+1)^7 - x^7 = 7x(x+1)(x^2+x+1)^2+1 (A121618). Values of x = A121619 - 1. Sequence of number of primes less than 10^n and of the form (x+1)^7 - x^7 have similar characteristics to similar sequences for natural primes (A006880), cuban primes (A113478) and primes of the form (x+1)^5 - x^5 (A221846).

Vladimir Pletser, Table of n, a(n) for n = 1..50

nn = 20; t = Table[0, {nn}]; n = 0; While[n++; p = (n + 1)^7 - n^7; p < 10^nn,If[PrimeQ[p], m = Ceiling[Log[10, p]]; t[[m]]++]]; Accumulate[t] (* T. D. Noe, Feb 04 2013 *)

A221978 Number of primes of the form (x+1)^7 - x^7 having n digits.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 1, 2, 4, 4, 7, 9, 12, 14, 29, 31, 45, 62, 71, 117, 175, 231, 331, 454, 634, 948, 1250, 1770, 2506, 3566, 5088, 7192, 10261, 14592, 21168, 30275, 43099, 61336, 87770, 126195, 180957, 258657, 371653, 534391, 767164, 1103259, 1583584, 2276179

Offset: 1

Vladimir Pletser, Feb 02 2013

Number of primes having n digits and equal to the difference of two consecutive seventh powers (x+1)^7 - x^7 = 7x(x+1)(x^2+x+1)^2+1 (A121618). Values of x = A121619 - 1. Sequence of number of primes having n digits and of the form (x+1)^7 - x^7 have similar characteristics to similar sequences for natural primes (A006879), cuban primes (A221792) and primes of the form (x+1)^5 - x^5 (A221847).

Vladimir Pletser, Table of n, a(n) for n = 1..50

nn = 30; t = Table[0, {nn}]; n = 0; While[n++; p = (n + 1)^7 - n^7; p < 10^nn, If[PrimeQ[p], m = Ceiling[Log[10, p]]; t[[m]]++]]; t (* T. D. Noe, Feb 04 2013 *)

A221979 Partial sums of primes of the form (n+1)^7 - n^7.

Original entry on oeis.org

127, 14324, 557931, 1831540, 4517357, 9734388, 26079025, 167982242, 2096276793, 10354981402, 24379848623, 47195272710, 78109546591, 169264277168, 285424955019, 468934979410, 749602296677, 1302535107108, 2819580695167, 4457920826414

Offset: 1

Vladimir Pletser, Feb 02 2013

Partial sums of primes equal to the difference of two consecutive seventh powers (x+1)^7 - x^7 = 7x(x+1)(x^2+x+1)^2+1 (A121618). Values of x = A121619 - 1. Number of primes equal (x+1)^7 - x^7 < 10^(n) in A221977. Partial sums of number of primes of the form (x+1)^7 - x^7 have similar characteristics to similar sequences for natural primes (A007504), cuban primes (A221793) and primes of the form (x+1)^5 - x^5 (A221848).

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

Accumulate[Select[Differences[Range[80]^7],PrimeQ]] (* Harvey P. Dale, Jul 09 2024 *)

A221980 Number of primes of the form (x+1)^7 - x^7 with x <= 10^n.

Original entry on oeis.org

1, 6, 24, 161, 1094, 8283, 66790

Offset: 0

Vladimir Pletser, Feb 02 2013

Number of primes equal to the difference of two consecutive seventh powers (x+1)^7 - x^7 = 7x(x+1)(x^2+x+1)^2+1 (A121618). Values of x = A121619 - 1. Sequence of number of primes of the form (x+1)^7 - x^7 with x <= 10^n have similar characteristics to similar sequences for natural primes, cuban primes (A221794) and primes of the form (x+1)^5 - x^5 (A221849).

Table[Count[#[[2]]-#[[1]]&/@Partition[Range[10^n+1]^7,2,1],?PrimeQ],{n,0,6}] (* _Harvey P. Dale, Sep 17 2019 *)

a(n) = {nb = 0; for (i = 1, 10^n, if (isprime((i+1)^7-i^7), nb++);); nb;} \\ Michel Marcus, Sep 18 2013

cp's OEIS Frontend

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

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

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A121620 Smallest prime of the form k^p - (k-1)^p, where p = prime(n).

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A121617 Numbers n such that A022521(n-1) = n^5 - (n-1)^5 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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A181126 Difference of two positive 7th powers.

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A221977 Number of primes of the form (x+1)^7 - x^7 less than 10^n.

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A221978 Number of primes of the form (x+1)^7 - x^7 having n digits.

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A221979 Partial sums of primes of the form (n+1)^7 - n^7.

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