A060859 Powerful numbers of the form k^2 - 1.
8, 288, 675, 9800, 235224, 332928, 1825200, 11309768, 384199200, 592192224, 4931691075, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 13325427460800, 15061377048200, 511643454094368
Offset: 1
Keywords
Examples
From _Jon E. Schoenfield_, Sep 06 2017: (Start) n k a(n) = k^2 - 1 a(n) + 1 = k^2 = === ========================= ================== 1 3 8 = 2^3 3^2 = 3^2 2 17 288 = 2^5 * 3^2 17^2 = 17^2 3 26 675 = 5^2 * 3^3 26^2 = 2^2 * 13^2 4 99 9800 = 2^3 * 5^2 * 7^2 99^2 = 3^4 * 11^2 5 485 235224 = 2^3 * 3^5 * 11^2 485^2 = 5^2 * 97^2 6 577 332928 = 2^7 * 3^2 * 17^2 577^2 = 577^2 (End)
Links
- Amiram Eldar, Table of n, a(n) for n = 1..55 (terms below 10^36; terms 1..30 from Donovan Johnson)
- Index entries for sequences related to powerful numbers.
Programs
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Mathematica
Select[Range[10^6]^2 - 1, Min[FactorInteger[#][[All, -1]]] > 1 &] (* Michael De Vlieger, Sep 05 2017 *) seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q, i}, q = Union[p, 2*Select[p, # <= max && OddQ[#] &]]; i = Position[Differences[q], 2] // Flatten; q[[i]]*(q[[i]] + 2)]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)
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PARI
isok(n) = issquare(n+1) && ispowerful(n); \\ Michel Marcus, Sep 05 2017
Formula
a(n) = k^2 - 1 and a(n) + 1 = k^2 are consecutive powerful numbers.
a(n) = A060860(n)^2 - 1. - Amiram Eldar, Feb 23 2024
Extensions
Corrected and extended by Jud McCranie, Jul 08 2001
Offset corrected by Donovan Johnson, Nov 15 2011
Name simplified by Jon E. Schoenfield, Nov 30 2023
Comments