A084371 -id:A084371 - cp's OEIS frontend

A365786 a(n) = squarefree kernel of A286708(n).

6, 6, 10, 6, 6, 14, 10, 6, 15, 6, 6, 14, 10, 6, 21, 22, 10, 6, 6, 15, 26, 14, 10, 6, 30, 22, 6, 10, 33, 15, 6, 34, 35, 6, 21, 26, 14, 38, 39, 14, 10, 6, 42, 30, 22, 6, 10, 15, 46, 6, 34, 10, 6, 51, 30, 26, 14, 38, 6, 55, 21, 14, 10, 57, 33, 58, 15, 6, 42, 30, 62

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Terms are squarefree and composite, i.e., in A120944.

			Let b(n) = A286708(n) and let squarefree kernel rad(n) = A007947(n).
a(1) = 6 = rad(b(1)) = rad(36).
a(2) = 6 = rad(b(2)) = rad(72).
a(3) = 10 = rad(b(3)) = rad(100), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A084371, A120944, A126706, A286708.

With[{nn = 2^12}, Map[Times @@ FactorInteger[#][[All, 1]] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not @* PrimePowerQ]] ]

apply(x->factorback(factorint(x)[, 1]), select(x->((x>1) && ispowerful(x) && !isprimepower(x)), [1..5000])) \\ Michel Marcus, Sep 20 2023

a(n) = A007947(A286708(n)).

A362972 Squarefree kernels of cubefull numbers (A036966).

Original entry on oeis.org

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

Amiram Eldar, May 13 2023

nonn

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

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.

Cf. A007947, A036966, A084371, A362974.

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]

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

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.

cp's OEIS Frontend

A365786 a(n) = squarefree kernel of A286708(n).

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