A188836 Numbers n for which A188794(n)^2 = n.

4, 9, 25, 49, 121, 169, 289, 361, 625, 841, 961, 1369, 1681, 1849, 3721, 4489, 5041, 5329, 7921, 9409, 10201, 10609, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 32761, 36481, 37249, 38809, 39601, 44521, 52441, 57121, 58081, 63001, 73441

Offset: 1

Author

Vladimir Shevelev, Apr 12 2011

Keywords

• nonn

Comments

The sequence contains many squares of primes.

Question 1: What is the sequence of primes whose squares are not in this sequence? It begins: 23, 47, 53, 59, 79, 83, 107, ... A188833

Question 2: What is the sequence of composite numbers whose squares are in this sequence? It begins: 25, 289, 361, 529, ...

Crossrefs

Cf. A188550, A188579, A188794, A188833.

Programs

Maple

with(numtheory):
b:= proc(n) local h, i, k, m;
       m, i:= 0, 0;
       for k from 2 to floor(sqrt(n)) do
          h:= nops(select(x-> irem(x, k)=0,
                  [seq (n-d, d=divisors(n-k) minus{1})]));
          if h>m then m, i:= h, k fi
       od; i
    end:
a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 3, a(n-1))
      while not b(k)^2=k do od; k
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Apr 13 2011

Mathematica

b[n_] := Module[{h, i = 0, k, m = 0}, For[k = 2, k <= Floor[Sqrt[n]], k++, h = Length[Select[Table[n - d, {d, Rest[Divisors[n - k]]}], Mod[#, k] == 0 &]]; If[h > m, {m, i} = {h, k}]]; i];
Reap[For[n = 1, n <= 80000, n++, If[b[n]^2==n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)