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%I #14 Sep 08 2022 08:45:54
%S 82,84,165,167,248,331,414,497,499,582,665,829,831,914,995,1080,1161,
%T 1246,1327,1329,1495,1576,1825,1910,2076,2157,2159,2323,2406,2408,
%U 2738,2821,2906,2989,3070,3238,3319,3485,3568,3651,3653,3817,4149,4234,4481
%N Numbers n such that 83 is the largest prime factor of n^2-1.
%C Sequence is finite, for proof see A175607.
%C Search for terms can be restricted to the range from 2 to A175607(23) = 34903240221563713; primepi(83) = 23.
%H A. Jasinski, <a href="/A181468/b181468.txt">Table of n, a(n) for n = 1..2082</a>
%t 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 < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 83, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==83&]
%o (Magma) [ n: n in [2..300000] | m eq 83 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 21 2011
%o (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..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 83 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 21 2011
%o (PARI) is(n)=n=n^2-1; forprime(p=2, 79, n/=p^valuation(n, p)); n>1 && 83^valuation(n, 83)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y Cf. A175607, A181447-A181467, A181469-A181470, A181568.
%K fini,nonn
%O 1,1
%A _Artur Jasinski_, Oct 21 2010
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