A120807 Cubes k in A120806: k+d+1 is prime for all divisors d of k. All cubes greater than 1 are cubes of odd primes.
1, 125, 357911, 28049850707778719, 1093838138707598549, 2498288375480240699, 2971816820123565959, 11368298790243739889, 14106863174732461979, 17104690428464397149, 21904077634699214681, 64352051556875937161, 82512915197756439761, 115892432166552995771, 231193025116112162501
Offset: 1
Keywords
Examples
a(3) = 357911 since k = 357911 = 71^3, divisors(k) = {1, 71, 71^2, 71^3} and k+d+1 = {357913, 357983, 362953, 715823} are all primes.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..2500
Programs
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Maple
L:=[]: for w to 1 do for k from 1 while nops(L)<=50 do p:=ithprime(k); x:=p^3; if p mod 6 = 5 and andmap(isprime,[x+2,2*x+1]) then S:={p,p^2}; Q:=map(z-> x+z+1, S); if andmap(isprime,Q) then L:=[op(L),x]; print(nops(L),p,x); fi; fi; od od; -
Mathematica
Select[Range[4008000]^3,AllTrue[#+Divisors[#]+1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 26 2019 *)
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PARI
lista(kmax) = {my(f, k3, is); forstep(k = 1, kmax, 2, f = factor(k); k3 = k^3; for(i = 1, #f~, f[i, 2] *= 3); is = 1; fordiv(f, d, if(!isprime(k3 + d + 1), is = 0; break)); if(is, print1(k3, ", ")));} \\ Amiram Eldar, Aug 05 2024
Extensions
a(13)-a(15) from Amiram Eldar, Aug 05 2024