A389340 Positive integers k = p_1^e_1*p_2^e_2*p_3^e_3, such that the points (p_1, e_1), (p_2, e_2) and (p_3, e_3) lie on a straight line with nonzero slope.

2160, 4725, 11250, 14553, 22000, 25725, 56000, 62073, 64800, 93296, 104949, 108864, 155520, 195657, 212625, 241893, 257125, 337500, 496125, 614169, 617584, 732050, 916839, 1047625, 1101373, 1146717, 1448128, 1530873, 1670625, 1795625, 1944000, 2117682, 2358125

Offset

1,1

Links

Felix Huber, Table of n, a(n) for n = 1..4298

Example

2160 = 2^4*3^3*5^1 is a term because (2, 4), (3, 3) and (5, 1) lie on a straight line.
11250 = 2^1*3^2*5^4 is a term because (2, 1), (3, 2) and (5, 4) lie on a straight line.
22000 = 2^4*5^3*11^1 is a term because (2, 4), (5, 3) and (11, 1) lie on a straight line.
64800 = 2^5*3^4*5^2 is a term because (2, 5), (3, 4) and (5, 2) lie on a straight line.

Maple

A389340:=proc(N) # To get all terms <= N
    local d, g, i, j, k, l, p, q, r, s, t;
    l:=[];
    for i while ithprime(i)^3*(ithprime(i)+1+i mod 2)^2*(ithprime(i)+3+i mod 2)<=N do
        p:=ithprime(i);
        for d while p^3*(p+d)^2*(p+2*d)<=N do
            d:=d+1
        od;
        d:=NumberTheory:-pi(p+2*d-2);
        for j from i+1 to d-1 do
            q:=ithprime(j);
            for k from j+1 to d do
                r:=ithprime(k);
                g:=gcd(q-p, r-q);
                s:=1;
                while p^(s*(r-p)/g+1)*q^(s*(r-q)/g+1)*r<=N do
                    t:=1;
                    while p^(s*(r-p)/g+t)*q^(s*(r-q)/g+t)*r^t<=N do
                        l:=[op(l), p^(s*(r-p)/g+t)*q^(s*(r-q)/g+t)*r^t];
                        t:=t+1
                    od;
                    s:=s+1;
                od;
                s:=1;
                while p*q^(s*(q-p)/g+1)*r^(s*(r-p)/g+1)<=N do
                    t:=1;
                    while p^t*q^(s*(q-p)/g+t)*r^(s*(r-p)/g+t)<=N do
                        l:=[op(l), p^t*q^(s*(q-p)/g+t)*r^(s*(r-p)/g+t)];
                        t:=t+1
                    od;
                    s:=s+1
                od
            od
        od
    od;
    return op(sort(l))
end proc;
A389340(2358125);

Crossrefs

Subsequence of A033992.

Cf. A338425, A373813, A389339.

Sequence in context: A152963 A179698 A075702 * A205629 A299796 A253715

Adjacent sequences: A389337 A389338 A389339 * A389341 A389342 A389343

Keyword

nonn

Author

Felix Huber, Oct 03 2025

Status

approved