A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 27625, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125

Offset

1,2

Comments

a(n) is least term of A054994 with exactly n divisors. - Ray Chandler, Jan 05 2012

From Jianing Song, Apr 24 2021: (Start)

a(n) is the smallest k such that A004018(k) = 4n.

Also a(n) is the smallest index of n in A002654.

a(n) is the smallest term in A004613 that has exactly n divisors.

This is a subsequence of A054994. (End)

Links

Ray Chandler, Table of n, a(n) for n = 1..1432 (a(1433) exceeds 1000 digits).

Formula

A000446(n) = min(a(2n-1), a(2n)) for n > 1.

A124980(n) = min(a(2n-1), a(2n)).

A016032(n) = min(2*a(2n-1), a(2n), a(2n+1)).

A093195(n) = min(a(2n), a(2n+1)).

From Jianing Song, Apr 24 2021: (Start)

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.

a(n) <= 5^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.

a(p*q) = 5^(p-1) * 13^(q-1) for primes p >= q. (End)

Example

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

Mathematica

(* 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 *)

Prog

(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
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
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.

Crossrefs

Cf. A002144, A002654, A004018, A005179, A054994, A000446, A124980, A016032, A093195.

Sequence in context: A007058 A071383 A088959 * A340388 A146665 A322594

Adjacent sequences: A018779 A018780 A018781 * A018783 A018784 A018785

Keyword

nonn

Author

N. J. A. Sloane

Status

approved