A343771 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).

1, 7, 49, 91, 2401, 637, 117649, 1729, 8281, 31213, 282475249, 12103, 13841287201, 1529437, 405769, 53599, 33232930569601, 157339, 1628413597910449, 593047, 19882681, 3672178237, 3909821048582988049, 375193, 68574961, 179936733613, 2989441, 29059303, 459986536544739960976801, 7709611

Offset

1,2

Comments

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

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

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

Links

Jianing Song, Table of n, a(n) for n = 1..1000

Helmut Ruhland, A family of lattices with an unbounded number of unit vectors, arXiv:2410.16172 [math.MG], 2024. See p. 2.

Formula

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]

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

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

Example

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.
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.

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
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.

Crossrefs

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

Analog of A005179 and A018782.

Sequence in context: A344473 A230655 A357302 * A008461 A043131 A082701

Adjacent sequences: A343768 A343769 A343770 * A343772 A343773 A343774

Keyword

nonn

Author

Jianing Song, Apr 28 2021

Status

approved