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A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.

%I A181452 #21 Dec 20 2024 18:15:31
%S A181452 16,33,35,50,67,69,101,103,118,120,169,188,239,271,307,339,441,511,
%T A181452 545,577,749,883,1121,1189,1376,1429,1665,1871,2024,2177,2311,2449,
%U A181452 2549,3401,4115,4861,4999,5201,9827,11663,24751,28799,57121,62425,74359,388961,672281
%N A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.
%C A181452 Numbers k such that A076605(k) = 17.
%C A181452 Sequence is finite, for proof see A175607.
%C A181452 Search for terms can be restricted to the range from 2 to A175607(7) = 672281; primepi(17) = 7.
%H A181452 Artur Jasinski, <a href="/A181452/b181452.txt">Table of n, a(n) for n = 1..47</a>
%t A181452 jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 700000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 17, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t A181452 Select[Range[680000], FactorInteger[#^2-1][[-1, 1]]==17&]
%o A181452 (Magma) [ n: n in [2..350000] | m eq 17 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 19 2011
%o A181452 (Magma) p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..700000] | p mod (n^2-1) eq 0 and (D[#D] eq 17 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 24 2011
%o A181452 (PARI) is(n)=n=n^2-1; forprime(p=2, 13, n/=p^valuation(n, p)); n>1 && 17^valuation(n, 17)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A181452 Cf. A076605, A175607, A181447-A181451, A181453-A181470, A181568.
%K A181452 fini,full,nonn
%O A181452 1,1
%A A181452 _Artur Jasinski_, Oct 21 2010