

A365983


Even numbers k such that k^2  1 is a powerful number.


0




OFFSET

1,1


COMMENTS

This sequence is a subsequence of A060860 (the even terms) and a supersequence of A094835. All the terms of A094835 are in this sequence, but 130576328 is not in A094835. A094835 also shows that this sequence is infinite.


REFERENCES

JeanMarie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, entries 70226 and 485.


LINKS



EXAMPLE

26^2  1 = 675 = 3^3 * 5^2 is powerful.
130576328^2  1 = 17050177433963583 = 3^2 * 7^3 * 13^2 * 293^2 * 617^2, whose exponents are all greater than 1, so it is powerful.


MATHEMATICA

seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3), 2}, {i, 1, Sqrt[max/j^3], 2}]]], i}, i = Position[Differences[p], 2] // Flatten; Sqrt[p[[i]]*(p[[i]] + 2) + 1]]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)


PROG

(PARI) isok(k) = !(k%2) && ispowerful(k^21); \\ Michel Marcus, Sep 25 2023


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



