A365983 Even numbers k such that k^2 - 1 is a powerful number.

26, 70226, 130576328, 189750626, 512706121226, 13837575261124, 99612037019890, 1385331749802026

Offset: 1

Author

Jud McCranie, Sep 24 2023

Keywords

• nonn
• more

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

Examples

			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.
		

References

• Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, entries 70226 and 485.

Links

• Index entries for sequences related to powerful numbers.

Crossrefs

Cf. A000466, A001694, A062739, A094835, A060860, A076445.

Programs

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 *)

PARI

isok(k) = !(k%2) && ispowerful(k^2-1); \\ Michel Marcus, Sep 25 2023

Extensions

a(5)-a(8) from Amiram Eldar, Feb 23 2024