A359231 -id:A359231 - cp's OEIS frontend

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

Ilya Gutkovskiy, Dec 22 2022

nonn

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)

			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.

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

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

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

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

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

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

cp's OEIS Frontend

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

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