A386600 a(n) is the number of primes q < p = prime(n) such that p - q is a perfect square.

0, 1, 0, 1, 2, 0, 1, 1, 2, 1, 0, 0, 2, 1, 3, 2, 2, 0, 2, 2, 1, 1, 5, 2, 1, 2, 2, 4, 1, 3, 0, 3, 3, 1, 2, 1, 1, 2, 6, 5, 2, 1, 2, 1, 4, 2, 1, 1, 8, 1, 4, 3, 1, 2, 5, 5, 3, 1, 1, 3, 1, 7, 2, 3, 1, 4, 1, 2, 5, 1, 6, 2, 3, 2, 0, 7, 3, 1, 3, 1, 4, 2, 4, 2, 0, 4, 3, 2

Offset

1,5

Comments

If prime(n) - 2 is a perfect square, then exactly one odd perfect square is included in the count. Otherwise, every perfect square counted is even.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Formula

a(A000720(A065377(n))) = 0.

a(A000720(A065376(n))) > 0.

Trivial upper bound: a(n) < A000720(n).

Example

a(5) = 2 because prime(5) = 11 and 11 - 2 = 3^2 and 11 - 7 = 2^2 are perfect squares. 11 - 3 = 8 and 11 - 5 = 6 are not perfect squares.

Maple

A386600:=proc(n)
    local a, i, p;
    a:=0;
    for i to n-1 do
        p:=ithprime(n);
        if issqr(p-ithprime(i)) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A386600(n), n=1..88);

Mathematica

a[n_]:=Total[Boole[IntegerQ/@Surd[Prime[n]-Prime[Range[n-1]], 2]]]; Array[a, 88] (* James C. McMahon, Oct 31 2025 *)

Prog

(PARI) a(n) = my(vp=primes(n)); sum(k=1, n-1, issquare(vp[n] - vp[k])); \\ Michel Marcus, Oct 27 2025

Crossrefs

Cf. A000040, A000290, A000720, A065376, A065377, A386603.

Sequence in context: A155041 A382545 A334296 * A227003 A307431 A263848

Adjacent sequences: A386597 A386598 A386599 * A386601 A386602 A386603

Keyword

nonn,easy

Author

Felix Huber, Oct 24 2025

Status

approved