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%I #55 Oct 08 2022 14:17:39
%S 96,98,193,195,290,389,484,581,583,775,872,874,969,971,1066,1163,1165,
%T 1359,1456,1551,1553,1648,1747,1844,1939,2036,2133,2135,2232,2521,
%U 2715,2911,3008,3103,3299,3394,3396,3590,3976,4267,4269,4463,4558,4946,5045
%N Numbers n such that 97 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(25) = 99913980938200001; primepi(97) = 25.
%H Lucas A. Brown, <a href="/A181470/b181470.txt">Table of n, a(n) for n = 1..2734</a> (2679 terms from Artur Jasinski, 55 missing terms from Lucas A. Brown, complete subject to the effective abc conjecture c < rad(abc)^2)
%H From _David A. Corneth_, Oct 03 2022: (Start)
%H To verify full I listed all 97-smooth numbers k that are a multiple of 97 below (inclusive) A175607(25) + 2. I then checked if k+2 is 97-smooth. If so, k+1 is a term. Then similarily I checked if k-2 is 97-smooth. If so, k-1 is a term.
%H Doing so found the 2734 terms from the b-file. All candidates have been checked completing the proof of full. (End)
%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 == 97, AppendTo[rr, n]]]; n++]; rr
%t (* or *)
%t Select[Range[300000], FactorInteger[#^2 - 1][[-1, 1]] == 97 &]
%o (Magma) [ n: n in [2..300000] | m eq 97 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 97 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 21 2011
%o (PARI) is(n)=n=n^2-1;forprime(p=2,89,n/=p^valuation(n,p));n>1 && 97^valuation(n,97)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y Cf. A175607, A181447-A181469, A181568.
%K nonn,fini,full
%O 1,1
%A _Artur Jasinski_, Oct 21 2010
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