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.

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A377045 Number of partitions of cuban primes.

Original entry on oeis.org

15, 490, 21637, 1121505, 3913864295, 1131238503938606, 78801255302666615, 5589233202595404488, 29349508915133986374841, 2163909235608484556362424, 913865816485680423486405066750, 191623400974625892978847721669762887224010
Offset: 1

Views

Author

Paul F. Marrero Romero, Oct 14 2024

Keywords

Comments

Number of partitions of prime numbers that are the difference of two consecutive cubes.
Number of partitions of primes p such that p=(3*k^2 + 1)/4 for some integer k (A121259).

Links

Crossrefs

Cf. A000041, A002407, A121259.

Programs

  • Maple
    R:= NULL: count:= 0:
for i from 1 while count < 30 do
  p:= (i+1)^3 - i^3;
  if isprime(p) then count:= count+1; v:= combinat:-numbpart(p); R:= R,v; fi
od:
R; # Robert Israel, Nov 14 2024
  • Mathematica
    PartitionsP[Select[Table[(3k^2 + 1)/4,{k,50}],PrimeQ]]

Formula

a(n) = A000041(A002407(n)).
a(n) = A000041((3*A121259(n)^2 + 1)/4).

A152217 Primes p == 1 (mod 3) such that ((p-1)/3)! == 1 (mod p).

Original entry on oeis.org

3571, 4219, 13669, 25117, 55897, 89269, 102121, 170647, 231019, 246247, 251431
Offset: 1

Views

Author

Francois Brunault (brunault(AT)gmail.com), Nov 29 2008, Nov 30 2008

Keywords

Comments

The Wilson theorem states that p is prime if and only if (p-1)! = -1 (mod p). If p = 3 (mod 4) then ((p-1)/2)! = +/- 1 (mod p).

Examples

			For n = 1 the prime a(1) = 3571 divides 1190! - 1.

References

  • J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.2.

Links

  • J. B. Cosgrave, Jacobi [From Francois Brunault (brunault(AT)gmail.com), Nov 29 2008]
  • Wikipedia, Wilson's theorem

Crossrefs

Seems to be a subsequence of A002407 and therefore of A003215 (differences of consecutive cubes). See also A058302 and A055939 for the sequences corresponding to ((p-1)/2)! = +/- 1 (mod p).

Programs

  • PARI
    forprime(p=2,30000,if(p%3==1 & ((p-1)/3)!%p==1,print(p)))

A210520 Number of cuban primes < 10^(n/2).

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 11, 17, 28, 42, 64, 105, 173, 267, 438, 726, 1200, 2015, 3325, 5524, 9289, 15659, 26494, 44946, 76483, 129930, 221530, 377856, 645685, 1105802, 1895983, 3254036, 5593440, 9625882, 16578830, 28590987, 49347768, 85253634
Offset: 0

Views

Author

Vladimir Pletser, Jan 26 2013

Keywords

Comments

A cuban prime has the form (x+1)^3 - x^3, which equals 3x*(x+1) + 1 (A002407).

Examples

			As the smallest cuban primes equal to the difference of two consecutive cubes p = (x+1)^3 - x^3, is 7 for x = 1, and as floor (10^(1/2)) = 3, a(0) = a(1) = 0 and a(2) = 1.

Crossrefs

Cf. A002407, A113478, A221794.

Programs

  • Mathematica
    cnt = 0; nxt = 1; t = {0}; Do[p = 3*k*(k + 1) + 1; If[p > nxt, AppendTo[t, cnt]; nxt = nxt*Sqrt[10]]; If[PrimeQ[p], cnt++], {k, 100000}]; t (* T. D. Noe, Jan 29 2013 *)
  • PARI
    b(n)={my(s=0,k=0,t=1); while(t<=n, s+=isprime(t); k++; t += 6*k); s}
a(n)={b(sqrtint(10^n))} \\ Andrew Howroyd, Jan 14 2020

Formula

a(2*n) = A113478(n). - Andrew Howroyd, Jan 14 2020

Extensions

a(31)-a(37) from Andrew Howroyd, Jan 14 2020
Previous Showing 31-33 of 33 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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