%I #46 May 08 2021 02:22:41
%S 1,5,25,65,625,325,15625,1105,4225,8125,9765625,5525,244140625,203125,
%T 105625,27625,152587890625,71825,3814697265625,138125,2640625,
%U 126953125,2384185791015625,160225,17850625,3173828125,1221025,3453125
%N Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.
%C a(n) is least term of A054994 with exactly n divisors. - _Ray Chandler_, Jan 05 2012
%C From _Jianing Song_, Apr 24 2021: (Start)
%C a(n) is the smallest k such that A004018(k) = 4n.
%C Also a(n) is the smallest index of n in A002654.
%C a(n) is the smallest term in A004613 that has exactly n divisors.
%C This is a subsequence of A054994. (End)
%H Ray Chandler, <a href="/A018782/b018782.txt">Table of n, a(n) for n = 1..1432</a> (a(1433) exceeds 1000 digits).
%F A000446(n) = min(a(2n-1), a(2n)) for n > 1.
%F A124980(n) = min(a(2n-1), a(2n)).
%F A016032(n) = min(2*a(2n-1), a(2n), a(2n+1)).
%F A093195(n) = min(a(2n), a(2n+1)).
%F From _Jianing Song_, Apr 24 2021: (Start)
%F If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 4. The smallest n such that the equality does not hold is n = 16.
%F a(n) <= 5^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
%F a(p*q) = 5^(p-1) * 13^(q-1) for primes p >= q. (End)
%e 4225 = 5^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 4 with exactly 9 divisors, so a(9) = 4225. - _Jianing Song_, Apr 24 2021
%t (* This program is not convenient to compute huge terms - A054994 is assumed to be computed with maxTerm = 10^16 *) a[n_] := Catch[ For[k = 1, k <= Length[A054994], k++, If[DivisorSigma[0, A054994[[k]]] == n, Throw[A054994[[k]]]]]]; Table[a[n], {n, 1, 28}] (* _Jean-François Alcover_, Jan 21 2013, after _Ray Chandler_ *)
%o (PARI) primelist(d,r,l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p,d)==r, i++; v[i]=p; if(i==l, break())))); v
%o prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
%o A018782(n)=my(pf=prodR(n, n), a=1, b, v=primelist(4, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(b<a || i==1, a=b)); a \\ _Jianing Song_, Apr 25 2021, following _R. J. Mathar_'s program for A005179.
%Y Cf. A002144, A002654, A004018, A005179, A054994, A000446, A124980, A016032, A093195.
%K nonn
%O 1,2
%A _N. J. A. Sloane_