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 1-6 of 6 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).

Links

Crossrefs

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

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 *)

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.

Links

Crossrefs

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

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 *)

A221846 Number of primes of the form (x+1)^5 - x^5 less than 10^n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 12, 19, 30, 40, 66, 110, 173, 285, 463, 749, 1256, 2075, 3499, 5884, 9928, 16754, 28345, 48037, 82187, 140358, 239768, 409315, 700510, 1200863, 2061093, 3544072, 6098353, 10505051
Offset: 0

Views

Author

Vladimir Pletser, Jan 26 2013

Keywords

Comments

Number of primes less than 10^n and equal to the difference of two consecutive fifth powers (x+1)^5 - x^5 = 5x(x+1)(x^2+x+1)+1 (A121616). Values of x = A121617. Sequence of number of primes less than 10^n and of the form (x+1)^5 - x^5 have similar characteristics to similar sequences for natural primes (A006880) and cuban primes (A113478).

A221848 Partial sums of primes equal (x+1)^5 - x^5.

Original entry on oeis.org

31, 242, 4893, 65944, 437225, 1161126, 2964127, 5825588, 10154739, 15080020, 22166471, 30110772, 44945803, 64557704, 87939735, 132059086, 186723797, 273272598, 371065129, 533543630, 723425661, 990642712, 1283752673, 1590492954, 2080592455
Offset: 1

Views

Author

Vladimir Pletser, Jan 26 2013

Keywords

Comments

Partial sums of primes equal to the difference of two consecutive fifth powers (x+1)^5 - x^5 = 5x(x+1)(x^2+x+1)+1 (A121616). Values of x = A121617. Number of primes equal (x+1)^5 - x^5 < 10^(n) in A221846. Partial sums of number of primes of the form (x+1)^5 - x^5 have similar characteristics to similar sequences for natural primes (A007504) and cuban primes (A221793).

Links

Programs

  • Mathematica
    Accumulate[Select[#[[2]]-#[[1]]&/@Partition[Range[100]^5,2,1],PrimeQ]] (* Harvey P. Dale, Mar 29 2013 *)

Extensions

More terms from Harvey P. Dale, Mar 29 2013

A221847 Number of primes of the form (x+1)^5 - x^5 having n digits.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 6, 7, 11, 10, 26, 44, 63, 112, 178, 286, 507, 819, 1424, 2385, 4044, 6826, 11591, 19692, 34150, 58171, 99410, 169547, 291195, 500353, 860230, 1482979, 2554281, 4406698
Offset: 0

Views

Author

Vladimir Pletser, Jan 26 2013

Keywords

Comments

Number of primes having n digits and equal to the difference of two consecutive fifth powers (x+1)^5 - x^5 = 5x(x+1)(x^2+x+1)+1 (A121616). Values of x = A121617. Sequence of number of primes having n digits and of the form (x+1)^5 - x^5 have similar characteristics to similar sequences for natural primes (A006879) and cuban primes (A221792).

A221849 Number of primes of the form (x+1)^5 - x^5 with x <= 10^n.

Original entry on oeis.org

1, 4, 25, 149, 1101, 8489, 69978, 596022, 5179467, 45811178
Offset: 0

Views

Author

Vladimir Pletser, Jan 26 2013

Keywords

Comments

Number of primes equal to the difference of two consecutive fifth powers (x+1)^5 - x^5 = 5x(x+1)(x^2+x+1)+1 (A121616) with x <= 10^n. Values of x = A121617. Sequence of number of primes of the form (x+1)^5 - x^5 with x <= 10^n have similar characteristics to similar sequences for natural primes and cuban primes (A221794).

Programs

  • Mathematica
    fQ[n_] := PrimeQ[(n + 1)^5 - n^5]; c = k = 0; Do[ While[k < 10^n + 1, If[ fQ@ k, c++]; k++]; Print[{n, c}], {n, 9}] (* Robert G. Wilson v, Jan 31 2013 *)

Extensions

a(7) - a(9) from Robert G. Wilson v, Jan 31 2013
Showing 1-6 of 6 results.

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