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

Author

Ilya Gutkovskiy, Dec 22 2022

Keywords

• nonn

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)

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Centered Square Number
• Index entries for sequences related to divisors of numbers

Crossrefs

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

Programs

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

Extensions

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

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