A389795 Numbers k with canonical prime factorization k = p1^e1*p2^e2*p3^e3*p4^e4 such that the points (p1, e1), (p2, e2), (p3, e3) and (p4, e4) form a square.

61689448894809898305, 153785952821337890625, 132323867879367231864225, 206059785628375473480829, 214482742566340717852669, 329870868801769775390625, 283834696601242712348762625, 707573013579796168212890625, 9517695438389034744606010681, 9906743396396711416896928441

Offset

1,1

Comments

The least term for which the square sidelength is an integer is a(1520) = 43^31*59^1*73^47*89^17 (sidelength 17).

Links

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

Example

61689448894809898305 = 3^9*5^1*11^11*13^3 is a term because the points (3, 9), (5, 1), (11, 11) and (13, 3) form a square (sidelength sqrt(68)).

Maple

A389795:=proc(N)
   local a, d, i, l, m, p, r, u, v, y;
   l:=[];
   for u while 2^2*3^(u+2)*(u+2)*(u+3)^(u+1)<=N do
      for v to u-1 while 2^(v+1)*(v+2)^(u+v+1)*(u+2)*(u+v+2)^(u+1)<=N do
         l:=[op(l), [[2, v+1], [v+2, u+v+1], [u+2, 1], [u+v+2, u+1]]]
      od
   od;
   for v while 2^(v+1)*3*(v+2)^(v+2)*(v+3)^2<=N do
      for u to v-1 while 2^(v+1)*(u+2)*(v+2)^(u+v+1)*(u+v+2)^(u+1)<=N do
         l:=[op(l), [[2, v+1], [u+2, 1], [v+2, u+v+1], [u+v+2, u+1]]]
      od
   od;
   a:=[];
   for i in l do
      r:=i;
      while r[1, 1]^r[1, 2]*r[2, 1]^r[2, 2]*r[3, 1]^r[3, 2]*r[4, 1]^r[4, 2]<=N do
         if isprime(r[2, 1]) and isprime(r[3, 1]) and isprime(r[4, 1]) then
            p:=r;
            m:=p[1, 1]^p[1, 2]*p[2, 1]^p[2, 2]*p[3, 1]^p[3, 2]*p[4, 1]^p[4, 2];
            for y from 0 while m<=N do
               a:=[op(a), m];
               p:=map(y->[y[1], y[2]+1], p);
               m:=p[1, 1]^p[1, 2]*p[2, 1]^p[2, 2]*p[3, 1]^p[3, 2]*p[4, 1]^p[4, 2]
            od;
         fi;
         d:=nextprime(r[1, 1])-i[1, 1];
         r:=map(x->[x[1]+d, x[2]], r)
      od
   od;
   return op(sort(a))
end proc;
A389795(9906743396396711416896928441);

Crossrefs

Subsequence of A033993.

Cf. A387172, A387586, A389339, A389340.

Sequence in context: A104283 A003852 A003845 * A144134 A172577 A276972

Adjacent sequences: A389792 A389793 A389794 * A389796 A389797 A389798

Keyword

nonn

Author

Felix Huber, Oct 17 2025

Status

approved