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A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.

%I A181453 #22 Dec 20 2024 18:15:43
%S A181453 18,20,37,39,56,77,113,134,151,153,170,191,246,265,305,324,341,362,
%T A181453 379,417,419,571,626,647,664,685,721,799,911,951,989,1025,1616,1937,
%U A181453 2431,2661,2889,3041,3079,3212,3457,3970,4751,4863,5851,6271,6499,8399,11551,11857
%N A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.
%C A181453 Numbers k such that A076605(k) = 19.
%C A181453 Sequence is finite, for proof see A175607.
%C A181453 Search for terms can be restricted to the range from 2 to A175607(8) = 23718421; primepi(19) = 8.
%H A181453 Artur Jasinski, <a href="/A181453/b181453.txt">Table of n, a(n) for n = 1..72</a>
%t A181453 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 < 24000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 19, AppendTo[rr, n]]]; n++ ]; rr
%t A181453 Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==19&]
%o A181453 (Magma) [ n: n in [2..300000] | m eq 19 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 18 2011
%o A181453 (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..24000000] | p mod (n^2-1) eq 0 and (D[#D] eq 19 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 24 2011
%o A181453 (PARI) is(n)=n=n^2-1; forprime(p=2, 17, n/=p^valuation(n, p)); n>1 && 19^valuation(n, 19)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A181453 Cf. A076605, A175607, A181447-A181452, A181454-A181470, A181568.
%K A181453 fini,full,nonn
%O A181453 1,1
%A A181453 _Artur Jasinski_, Oct 21 2010