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%I #18 Sep 08 2022 08:45:54
%S 52,54,105,107,160,211,319,370,476,529,531,584,637,741,743,847,849,
%T 900,902,953,1059,1220,1273,1324,1377,1379,1483,1538,1644,1695,1801,
%U 1803,2015,2174,2278,2386,2437,2543,2651,2755,2861,2969,3073,3181,3497,3499
%N Numbers n such that 53 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(16) = 2907159732049; primepi(53) = 16.
%H A. Jasinski, <a href="/A181461/b181461.txt">Table of n, a(n) for n = 1..507</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 == 53, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==53&]
%o (Magma) [ n: n in [2..300000] | m eq 53 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 19 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 53 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 20 2011
%o (PARI) is(n)=n=n^2-1; forprime(p=2, 47, n/=p^valuation(n, p)); n>1 && 53^valuation(n, 53)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y Cf. A175607, A181447-A181460, A181462-A181470, A181568.
%K fini,nonn
%O 1,1
%A _Artur Jasinski_, Oct 21 2010
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