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A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.

%I A181459 #22 Dec 20 2024 18:16:46
%S A181459 42,44,85,87,171,173,216,257,259,300,343,386,431,474,517,560,601,687,
%T A181459 689,730,818,859,1074,1117,1119,1289,1291,1332,1420,1549,1633,1721,
%U A181459 1805,1891,1977,1979,2108,2321,2495,2665,2667,2751,2753,2794,2925,3095,3484
%N A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.
%C A181459 Numbers k such that A076605(k) = 43.
%C A181459 Sequence is finite, for proof see A175607.
%C A181459 Search for terms can be restricted to the range from 2 to A175607(14) = 842277599279; primepi(43) = 14.
%H A181459 Artur Jasinski, <a href="/A181459/b181459.txt">Table of n, a(n) for n = 1..343</a>
%t A181459 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 == 43, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t A181459 Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==43&]
%o A181459 (Magma) [ n: n in [2..300000] | m eq 43 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 19 2011
%o A181459 (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 43 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 20 2011
%o A181459 (PARI) is(n)=n=n^2-1; forprime(p=2, 41, n/=p^valuation(n, p)); n>1 && 43^valuation(n, 43)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A181459 Cf. A076605, A175607, A181447-A181458, A181460-A181470, A181568.
%K A181459 fini,full,nonn
%O A181459 1,1
%A A181459 _Artur Jasinski_, Oct 21 2010