cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Original entry on oeis.org

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

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Author

Alexander Adamchuk, Aug 10 2006

Keywords

Comments

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).

Crossrefs

Programs

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

Views

Author

Alexander Adamchuk, Aug 10 2006

Keywords

Comments

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,...}.

Crossrefs

Programs

Extensions

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

Views

Author

Alexander Adamchuk, Aug 10 2006

Keywords

Comments

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.

Crossrefs

Programs

  • Mathematica
    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

Views

Author

Alexander Adamchuk, Aug 10 2006

Keywords

Comments

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, ...}.

Crossrefs

Programs

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

Views

Author

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

Keywords

Comments

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.

Crossrefs

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.

Formula

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

Views

Author

T. D. Noe, Oct 06 2010

Keywords

Comments

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?

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Vladimir Pletser, Feb 02 2013

Keywords

Comments

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).

Programs

  • Mathematica
    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

Views

Author

Vladimir Pletser, Feb 02 2013

Keywords

Comments

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).

Programs

  • Mathematica
    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

Views

Author

Vladimir Pletser, Feb 02 2013

Keywords

Comments

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).

Programs

  • Mathematica
    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

Views

Author

Vladimir Pletser, Feb 02 2013

Keywords

Comments

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).

Programs

Showing 1-10 of 10 results.