A378896 -id:A378896 - cp's OEIS frontend

A379018 a(n) is the least prime p such that p^2 < n and n - p^2 is not squarefree, or -1 if there is no such p.

-1, -1, -1, -1, -1, -1, -1, 2, -1, -1, -1, 2, 2, -1, -1, 2, 3, 3, -1, 2, 3, 2, -1, 2, 3, -1, 3, 2, 2, -1, 2, 2, 3, 3, -1, 2, 3, -1, -1, 2, 3, -1, 5, 2, 3, -1, -1, 2, 2, 5, -1, 2, 2, 2, -1, 2, 3, 2, 3, 2, 3, -1, 3, 2, 3, -1, 2, 2, 3, 5, -1, 2, 3, 5, 5, 2, 3, -1, 2, 2, 3, -1, -1, 2, 2, -1, -1, 2, 3

Offset: 1

Robert Israel, Dec 13 2024

sign

a(n) is the least p, if it exists, such that there is a solution to the equation n = p^2 + k * q^2 with p and q prime and k > 0.

			a(17) = 3 because 17 - 3^2 = 8 is not squarefree, while 17 - 2^2 = 13 is squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A013929, A378896.

f:= proc(n) local p;
  p:= 2;
  while p^2 <= n do
    if not numtheory:-issqrfree(n-p^2) then return p fi;
    p:= nextprime(p);
  od;
  -1
end proc:
map(f, [$1..200]);

A377842 a(n) = q - 2p, where q is the greatest prime such that p=2n - q is also prime.

Original entry on oeis.org

-2, -3, -1, 1, -3, 5, 7, 3, 11, 13, 9, 17, 13, 9, 23, 25, 21, 17, 31, 27, 35, 37, 33, 41, 37, 33, 47, 43, 39, 53, 55, 51, 47, 61, 57, 65, 67, 63, 59, 73, 69, 77, 73, 69, 83, 79, 75, 41, 91, 87, 95, 97, 93, 101, 103, 99, 107, 103, 99, 83, 91, 87, 71, 121, 117, 125, 121, 117, 131, 133

Offset: 2

Michel Eduardo Beleza Yamagishi, Dec 10 2024

sign

M. E. B. Yamagishi, Goldbach's Conjecture: A Simple and Fast Algorithm, Preprint, 2024.

Cf. A020481, A020482, A378896.

Table[Module[{p = 2, q},
  While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
   p = NextPrime[p]]; q - 2 p], {n, 2, 100}]

a(n) = my(q=precprime(2*n)); while (!isprime(2*n - q), q = precprime(q-1)); q - 2*(2*n-q); \\ Michel Marcus, Dec 12 2024

cp's OEIS Frontend

A379018 a(n) is the least prime p such that p^2 < n and n - p^2 is not squarefree, or -1 if there is no such p.

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A377842 a(n) = q - 2p, where q is the greatest prime such that p=2n - q is also prime.

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PARI

cp's OEIS Frontend

A379018 a(n) is the least prime p such that p^2 < n and n - p^2 is not squarefree, or -1 if there is no such p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A377842 a(n) = q - 2*p, where q is the greatest prime such that p=2*n - q is also prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A377842 a(n) = q - 2p, where q is the greatest prime such that p=2n - q is also prime.