A387172 Numbers k = p1^e1*p2^e2, with e1 != e2, such that the Euclidean distance between points (p1, e1) and (p2, e2) is an integer.

160, 567, 1600, 6250, 7203, 8704, 11907, 16000, 24057, 26411, 57344, 62500, 90112, 102487, 151263, 160000, 250047, 265625, 295936, 303104, 485537, 625000, 793881, 802816, 1015625, 1085773, 1600000, 1982464, 2033647, 2235331, 2539107, 2997383, 3176523, 5250987, 5316979

Offset

1,1

Links

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

Formula

If t is a term, then t*(p_1*p_2)^m = t*A007947(t)^m is also a term for any positive integer m.

Example

160 = 2^5*5^1 has exactly two prime-power factors with distinct exponents and the lattice distance between (2, 5) and (5, 1) is an integer: sqrt((5 - 2)^2 + (1 - 5)^2) = 5. Hence, 160 is a term.

Maple

A387172:=proc(N) # To get all terms <= N
    local p, i, j, k, l, m, x, y, u, v;
    l:=[];
    for x from 2 while min(2^(2*x+1)*(x^2+1), 2^(x^2)*(2*x+2))<=N do
        for y to x-1 do
            if gcd(x, y)=1 and is(x-y, odd) then
                u:=x^2-y^2;
                v:=2*x*y;
                for i while min(2^(u*i+1)*(v*i+2), 2^(v*i+1)*(u*i+2))<=N do
                    l:=[op(l), i*[v, u]];
                    l:=[op(l), i*[u, v]];
                od
            fi
        od
    od;
    m:=[];
    for k in l do
        for i while ithprime(i)^(k[1]+1)*(ithprime(i)+k[2])<=N do
            p:=ithprime(i);
            for j while p^(k[1]+j)*(p+k[2])^j<=N do
                if isprime(p+k[2]) then
                    m:=[op(m), p^(k[1]+j)*(p+k[2])^j];
                    if p^j*(p+k[2])^(k[1]+j)<=N then
                        m:=[op(m), p^j*(p+k[2])^(k[1]+j)];
                    fi
                fi
            od
        od
    od;
    return op(sort(m))
end proc;
A387172(5316979);

Crossrefs

Subsequence of A007774.

Cf. A007947, A389339, A389340.

Sequence in context: A133530 A278129 A184070 * A278122 A233917 A233910

Adjacent sequences: A387169 A387170 A387171 * A387173 A387174 A387175

Keyword

nonn

Author

Felix Huber, Oct 06 2025

Status

approved