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Powerful numbers of the form k^2 - 1.
5

%I #43 Feb 23 2024 20:12:19

%S 8,288,675,9800,235224,332928,1825200,11309768,384199200,592192224,

%T 4931691075,13051463048,221322261600,443365544448,865363202000,

%U 8192480787000,13325427460800,15061377048200,511643454094368

%N Powerful numbers of the form k^2 - 1.

%C If k^2-1 is a term, then k-1 is a term of A335851. - _Amiram Eldar_, Feb 23 2024

%H Amiram Eldar, <a href="/A060859/b060859.txt">Table of n, a(n) for n = 1..55</a> (terms below 10^36; terms 1..30 from Donovan Johnson)

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F a(n) = k^2 - 1 and a(n) + 1 = k^2 are consecutive powerful numbers.

%F a(n) = A060860(n)^2 - 1. - _Amiram Eldar_, Feb 23 2024

%e From _Jon E. Schoenfield_, Sep 06 2017: (Start)

%e n k a(n) = k^2 - 1 a(n) + 1 = k^2

%e = === ========================= ==================

%e 1 3 8 = 2^3 3^2 = 3^2

%e 2 17 288 = 2^5 * 3^2 17^2 = 17^2

%e 3 26 675 = 5^2 * 3^3 26^2 = 2^2 * 13^2

%e 4 99 9800 = 2^3 * 5^2 * 7^2 99^2 = 3^4 * 11^2

%e 5 485 235224 = 2^3 * 3^5 * 11^2 485^2 = 5^2 * 97^2

%e 6 577 332928 = 2^7 * 3^2 * 17^2 577^2 = 577^2

%e (End)

%t Select[Range[10^6]^2 - 1, Min[FactorInteger[#][[All, -1]]] > 1 &] (* _Michael De Vlieger_, Sep 05 2017 *)

%t 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 *)

%o (PARI) isok(n) = issquare(n+1) && ispowerful(n); \\ _Michel Marcus_, Sep 05 2017

%Y Proper subset of A060355.

%Y Cf. A001694, A060860, A335851.

%K nonn

%O 1,1

%A _Labos Elemer_, May 04 2001

%E Corrected and extended by _Jud McCranie_, Jul 08 2001

%E Offset corrected by _Donovan Johnson_, Nov 15 2011

%E Name simplified by _Jon E. Schoenfield_, Nov 30 2023