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A181458 Numbers k such that 41 is the largest prime factor of k^2 - 1.

%I A181458 #21 Dec 20 2024 18:16:38
%S A181458 40,81,83,122,124,163,204,206,247,286,288,329,409,491,493,573,575,737,
%T A181458 739,778,901,944,985,1024,1065,1106,1149,1231,1393,1518,1559,1639,
%U A181458 1682,2049,2051,2092,2295,2377,2379,2623,2705,2789,3035,3158,3199,3361,3363
%N A181458 Numbers k such that 41 is the largest prime factor of k^2 - 1.
%C A181458 Numbers k such that A076605(k) = 41.
%C A181458 Sequence is finite, for proof see A175607.
%C A181458 Search for terms can be restricted to the range from 2 to A175607(13) = 127855050751; primepi(41) = 13.
%H A181458 Artur Jasinski, <a href="/A181458/b181458.txt">Table of n, a(n) for n = 1..262</a>
%t A181458 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 == 41, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t A181458 Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==41&]
%o A181458 (Magma) [ n: n in [2..300000] | m eq 41 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 19 2011
%o A181458 (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 41 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 20 2011
%o A181458 (PARI) is(n)=n=n^2-1; forprime(p=2, 37, n/=p^valuation(n, p)); n>1 && 41^valuation(n, 41)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A181458 Cf. A076605, A175607, A181447-A181457, A181459-A181470, A181568.
%K A181458 fini,full,nonn
%O A181458 1,1
%A A181458 _Artur Jasinski_, Oct 21 2010