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A365983 Even numbers k such that k^2 - 1 is a powerful number. 0
26, 70226, 130576328, 189750626, 512706121226, 13837575261124, 99612037019890, 1385331749802026 (list; graph; refs; listen; history; text; internal format)
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.
Terms A076445(n)+1 are terms of this sequence because A076445(n) and A076445(n)+2 are powerful and A076445(n)+1)^2-1 = A076445(n) * (A076445(n)+2), which is also powerful.
a(n) - 1 is an odd powerful number (A062739). - Amiram Eldar, Feb 23 2024
REFERENCES
Jean-Marie 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^2-1); \\ Michel Marcus, Sep 25 2023
CROSSREFS
Sequence in context: A208186 A316677 A092212 * A265959 A112946 A034246
KEYWORD
nonn,more
AUTHOR
Jud McCranie, Sep 24 2023
EXTENSIONS
a(5)-a(8) from Amiram Eldar, Feb 23 2024
STATUS
approved

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Last modified May 19 10:22 EDT 2024. Contains 372683 sequences. (Running on oeis4.)