A359232 a(n) is the smallest centered square number divisible by exactly n centered square numbers.

1, 5, 25, 925, 1625, 1105, 47125, 350285, 493025, 3572465, 47074105, 13818025, 4109345825, 171921425, 294346585, 130334225125, 190608050165, 2687125303525, 2406144489125, 5821530534625, 49723952067725, 1500939251825, 665571884367325, 8362509238504525, 1344402738869125

Offset

1,2

Comments

From Jon E. Schoenfield, Dec 24 2022: (Start)

For all n > 22, a(n) > 5*10^14.

For all n in 10..22, the prime factors of a(n) include 5, 13, and 17. Every index k such that 5*13*17=1105 divides the k-th centered square number satisfies k == { 23, 231, 418, 431, 673, 686, 873, 1081 } (mod 1105), so a search for upper bounds for larger terms can be facilitated by testing only such indices k.

Some known upper bounds: a(23) <= 665571884367325, a(24) <= 8362509238504525, a(25) <= 1344402738869125, a(26) <= 49165090920807485, a(27) <= 4384711086003625, a(30) <= 13148945184367525, a(33) <= 179899779754020625. (End)

Links

Table of n, a(n) for n=1..25.

Eric Weisstein's World of Mathematics, Centered Square Number

Index entries for sequences related to divisors of numbers

Example

a(5) = 1625, because 1625 is a centered square number that has 5 centered square divisors {1, 5, 13, 25, 1625} and this is the smallest such number.

Prog

(Magma)
a := [ 0 : n in [ 1 .. 17 ] ];
for k in [ 0 .. 310000 ] do
   c := 2*k*(k+1)+1;
   D := Divisors(c);
   n := 0;
   for d in D do
      if IsSquare(2*d - 1) then
         n +:= 1;
      end if;
   end for;
   if a[n] eq 0 then
      a[n] := c;
   end if;
end for;
a; // Jon E. Schoenfield, Dec 24 2022
(PARI) a(n) = for(k=0, oo, my(t=2*k*(k+1)+1); if(sumdiv(t, d, issquare(2*d-1)) == n, return(t))); \\ Daniel Suteu, Dec 31 2022

Crossrefs

Cf. A001844, A130279, A300410, A358545, A358861, A359231.

Sequence in context: A274463 A279835 A169652 * A359235 A137114 A067212

Adjacent sequences: A359229 A359230 A359231 * A359233 A359234 A359235

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 22 2022

Extensions

a(10)-a(22) from Jon E. Schoenfield, Dec 24 2022

a(23)-a(25) confirmed by Daniel Suteu, Dec 31 2022

Status

approved