A362972 Squarefree kernels of cubefull numbers (A036966).

1, 2, 2, 3, 2, 2, 3, 5, 2, 6, 3, 2, 7, 6, 2, 5, 6, 3, 6, 10, 2, 6, 11, 6, 6, 10, 2, 3, 13, 7, 6, 14, 5, 15, 6, 6, 10, 2, 17, 10, 6, 14, 6, 3, 19, 6, 6, 10, 2, 21, 10, 15, 6, 22, 14, 6, 23, 6, 11, 6, 5, 10, 2, 7, 15, 6, 26, 14, 3, 10, 6, 22, 14, 6, 29, 10, 30, 6

Offset

1,2

Links

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

Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, eq. (18), p. 728.

Formula

a(n) = A007947(A036966(n)).

Sum_{A036966(k) < x} a(k) = c * x^(2/3) + o(x^(2/3)), where c = (3/Pi^2) * Product_{p prime} (1 + 1/((p+1)*(p^(2/3)-1)) = 0.7356919531... (Jakimczuk, 2017). [corrected Sep 21 2024]

Sum_{k=1..n} a(k) ~ (c / A362974 ^ 2) * n^2, where c is the constant above.

Example

A036966(2) = 8 = 2^3, therefore a(2) = 2.
A036966(10) = 216 = 2^3 * 3^2, therefore a(10) = 2 * 3 = 6.

Mathematica

seq[kmax_] := Module[{s = {1}}, Do[f = FactorInteger[k]; If[Min@f[[;; , 2]] > 2, AppendTo[s, Times @@ f[[;; , 1]]]], {k, 2, kmax}]; s]; seq[10^5]

Prog

(PARI) lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(k==1 || vecmin(f[, 2]) > 2, print1(vecprod(f[, 1]), ", "))); }

Crossrefs

Cf. A007947, A036966, A084371, A362974.

Sequence in context: A258568 A108939 A370835 * A138143 A285727 A281908

Adjacent sequences: A362969 A362970 A362971 * A362973 A362974 A362975

Keyword

nonn,changed

Author

Amiram Eldar, May 13 2023

Status

approved