A399522 - OEIS

A399522

Indices of the Achilles numbers (A052486) in the sequence of powerful numbers (A001694).

12, 15, 22, 27, 32, 34, 37, 42, 43, 47, 49, 52, 53, 57, 58, 62, 64, 66, 69, 74, 77, 78, 86, 89, 91, 95, 98, 101, 103, 106, 108, 112, 113, 115, 118, 121, 122, 123, 128, 131, 132, 134, 135, 137, 142, 144, 146, 150, 153, 155, 159, 162, 169, 171, 172, 173, 175, 178, 182, 183

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OFFSET

1,1

COMMENTS

Equivalently, the number of powerful numbers that do not exceed the n-th Achilles number.

The asymptotic density of this sequence is 1 - zeta(3)/zeta(3/2) = 1 - 1/A090699 = 0.539860570279... .

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

FORMULA

a(n) = A217038(A052486(n)) + 1.

A001694(a(n)) = A052486(n).

a(n) = c * n + O(n^(2/3)), where c = 1/(1 - zeta(3)/zeta(3/2)) = 1.8523301... .

EXAMPLE

The first 12 powerful numbers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, and 72. 72 is the first Achilles number. Therefore, a(1) = 12.

MATHEMATICA

achillesQ[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 1 &] && GCD @@ e == 1];

seq[lim_] := Position[Union[Table[i^2*j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}] // Flatten], _Integer?achillesQ] // Flatten;

seq[10^4]

PROG

(PARI) isAchilles(n) = if(n == 1, 0, my(e = factor(n)[, 2]); vecmin(e) > 1 && gcd(e) == 1);

list(lim) = {my(v = List()); for(j = 1, sqrtnint(lim\1, 3), for(i = 1, sqrtint(lim\j^3), listput(v, i^2*j^3))); Vec(select(x -> isAchilles(x), Set(v), 1)); }

CROSSREFS

Cf. A001694, A052486, A090699, A217038, A399524, A399525.

Similar sequences: A361936, A371185, A371186, A399523 (complement).

Sequence in context: A259040 A158190 A122040 * A342758 A274550 A253235

Adjacent sequences: A399505 A399506 A399507 * A399523 A399524 A399525

KEYWORD

nonn,easy,new

AUTHOR

Amiram Eldar, Sep 01 2026

STATUS

approved