A343771 - OEIS

%I #25 Oct 25 2024 17:06:24

%S 1,7,49,91,2401,637,117649,1729,8281,31213,282475249,12103,

%T 13841287201,1529437,405769,53599,33232930569601,157339,

%U 1628413597910449,593047,19882681,3672178237,3909821048582988049,375193,68574961,179936733613,2989441,29059303,459986536544739960976801,7709611

%N Smallest k such that circle centered at the origin and with radius sqrt(k) passes through exactly 6*n integer points in the hexagonal lattice (see A004016).

%C a(n) is the smallest k such that A004016(k) = 6*n.

%C Also a(n) is the smallest index of n in A002324.

%C a(n) is the smallest term in A004611 that has exactly n divisors.

%H Jianing Song, <a href="/A343771/b343771.txt">Table of n, a(n) for n = 1..1000</a>

%H Helmut Ruhland, <a href="https://arxiv.org/abs/2410.16172">A family of lattices with an unbounded number of unit vectors</a>, arXiv:2410.16172 [math.MG], 2024. See p. 2.

%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 3. The smallest n such that the equality does not hold is n = 128. [Those n such that the equality does not hold are listed in A355919. - _Jianing Song_, Jul 21 2022]

%F a(n) <= 7^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.

%F a(p*q) = 7^(p-1) * 13^(q-1) for primes p >= q.

%e 91 = 7 * 13 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 4 divisors, so a(4) = 91.

%e 8281 = 7^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 9 divisors, so a(9) = 8281.

%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 A343771(n)=my(pf=prodR(n, n), a=1, b, v=primelist(3, 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 \\ following program for A005179.

%Y Cf. A004016, A002324, A004611, A230655, A355919.

%Y Analog of A005179 and A018782.

%K nonn

%O 1,2

%A _Jianing Song_, Apr 28 2021