cp's OEIS Frontend

A060859 Powerful numbers of the form k^2 - 1.

%I A060859 #43 Feb 23 2024 20:12:19
%S A060859 8,288,675,9800,235224,332928,1825200,11309768,384199200,592192224,
%T A060859 4931691075,13051463048,221322261600,443365544448,865363202000,
%U A060859 8192480787000,13325427460800,15061377048200,511643454094368
%N A060859 Powerful numbers of the form k^2 - 1.
%C A060859 If k^2-1 is a term, then k-1 is a term of A335851. - _Amiram Eldar_, Feb 23 2024
%H A060859 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 A060859 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A060859 a(n) = k^2 - 1 and a(n) + 1 = k^2 are consecutive powerful numbers.
%F A060859 a(n) = A060860(n)^2 - 1. - _Amiram Eldar_, Feb 23 2024
%e A060859 From _Jon E. Schoenfield_, Sep 06 2017: (Start)
%e A060859 n     k        a(n) =  k^2 - 1          a(n) + 1 = k^2
%e A060859 =   ===   =========================   ==================
%e A060859 1     3        8 = 2^3                  3^2 = 3^2
%e A060859 2    17      288 = 2^5 * 3^2           17^2 = 17^2
%e A060859 3    26      675 = 5^2 * 3^3           26^2 = 2^2 * 13^2
%e A060859 4    99     9800 = 2^3 * 5^2 * 7^2     99^2 = 3^4 * 11^2
%e A060859 5   485   235224 = 2^3 * 3^5 * 11^2   485^2 = 5^2 * 97^2
%e A060859 6   577   332928 = 2^7 * 3^2 * 17^2   577^2 = 577^2
%e A060859 (End)
%t A060859 Select[Range[10^6]^2 - 1, Min[FactorInteger[#][[All, -1]]] > 1 &] (* _Michael De Vlieger_, Sep 05 2017 *)
%t A060859 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 A060859 (PARI) isok(n) = issquare(n+1) && ispowerful(n); \\ _Michel Marcus_, Sep 05 2017
%Y A060859 Proper subset of A060355.
%Y A060859 Cf. A001694, A060860, A335851.
%K A060859 nonn
%O A060859 1,1
%A A060859 _Labos Elemer_, May 04 2001
%E A060859 Corrected and extended by _Jud McCranie_, Jul 08 2001
%E A060859 Offset corrected by _Donovan Johnson_, Nov 15 2011
%E A060859 Name simplified by _Jon E. Schoenfield_, Nov 30 2023