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

%I A359232 #16 Feb 16 2025 08:34:04
%S A359232 1,5,25,925,1625,1105,47125,350285,493025,3572465,47074105,13818025,
%T A359232 4109345825,171921425,294346585,130334225125,190608050165,
%U A359232 2687125303525,2406144489125,5821530534625,49723952067725,1500939251825,665571884367325,8362509238504525,1344402738869125
%N A359232 a(n) is the smallest centered square number divisible by exactly n centered square numbers.
%C A359232 From _Jon E. Schoenfield_, Dec 24 2022: (Start)
%C A359232 For all n > 22, a(n) > 5*10^14.
%C A359232 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.
%C A359232 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)
%H A359232 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>
%H A359232 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A359232 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.
%o A359232 (Magma)
%o A359232 a := [ 0 : n in [ 1 .. 17 ] ];
%o A359232 for k in [ 0 .. 310000 ] do
%o A359232    c := 2*k*(k+1)+1;
%o A359232    D := Divisors(c);
%o A359232    n := 0;
%o A359232    for d in D do
%o A359232       if IsSquare(2*d - 1) then
%o A359232          n +:= 1;
%o A359232       end if;
%o A359232    end for;
%o A359232    if a[n] eq 0 then
%o A359232       a[n] := c;
%o A359232    end if;
%o A359232 end for;
%o A359232 a; // _Jon E. Schoenfield_, Dec 24 2022
%o A359232 (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
%Y A359232 Cf. A001844, A130279, A300410, A358545, A358861, A359231.
%K A359232 nonn
%O A359232 1,2
%A A359232 _Ilya Gutkovskiy_, Dec 22 2022
%E A359232 a(10)-a(22) from _Jon E. Schoenfield_, Dec 24 2022
%E A359232 a(23)-a(25) confirmed by _Daniel Suteu_, Dec 31 2022