

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 nth divisor of a number k, for n>1; and a(1) = 1.


0



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

1,2


COMMENTS

Or a(n) is the smallest number k such that, if d is the nth 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



