A286527 a(n) is the smallest k>1 such that d(n,k)^2 = d(n^2,k^2), where d(n,k) is the n-th divisor of a number k, for n>1; and a(1) = 1.

1, 6, 70, 210, 2622, 9282, 277134, 1159710, 8064030, 56185590, 186605430, 2748628830, 5053814978, 72641163166

Offset

1,2

Comments

Or a(n) is the smallest number k such that, if d is the n-th divisor of k, then d^2 is the (n^2)-th divisor of k^2.

For n <= 14, a(n) is squarefree, and omega(a(n)) < 9. Is a(n) squarefree for all n? - David A. Corneth, May 12 2017

Links

Table of n, a(n) for n=1..14.

Example

a(2) = 6 because the divisors of 6 and 36 are {1, 2, 3, 6} and {1, 2, 3, 4, 6, 9, 12, 18, 36} respectively, and the 2nd divisor of 6 is 2, and the 4th divisor of 36 is 2^2. Hence, d(2,6)^2 = d(4,36) = 4.

Mathematica

Do[k=1; While[!(Length[Divisors[k]]>=n&&Length[Divisors[k^2]]>=n^2&&Part[Divisors[k], n]^2==Part[Divisors[k^2], n^2]), k++]; Print[n, " ", k], {n, 1, 10}]

Prog

(PARI) a(n) = {if (n==1, return (1)); my(k=2); while (iferr(divisors(k)[n]^2 != divisors(k^2)[n^2], E, 1), k++); k; } \\ Michel Marcus, Sep 12 2017

Crossrefs

Cf. A027750.

Sequence in context: A218683 A188406 A048708 * A104900 A186667 A001448

Adjacent sequences: A286524 A286525 A286526 * A286528 A286529 A286530

Keyword

nonn,more

Author

Michel Lagneau, May 11 2017

Extensions

a(10)-a(12) from Giovanni Resta, May 12 2017

a(13)-a(14) from Giovanni Resta, May 16 2017

Name edited by Michel Marcus, Sep 15 2017

Status

approved