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A181467 Numbers k such that 79 is the largest prime factor of k^2 - 1.

%I A181467 #18 Dec 21 2024 17:59:08
%S A181467 78,80,157,159,236,317,473,475,552,554,631,712,791,868,870,947,949,
%T A181467 1026,1028,1105,1184,1421,1737,1739,1816,1897,2053,2134,2211,2213,
%U A181467 2369,2450,2529,2685,2687,2843,2924,3001,3161,3477,3554,3870,3949,3951,4186,4188
%N A181467 Numbers k such that 79 is the largest prime factor of k^2 - 1.
%C A181467 Numbers k such that A076605(k) = 79.
%C A181467 Sequence is finite, for proof see A175607.
%C A181467 Search for terms can be restricted to the range from 2 to A175607(22) = 19182937474703818751; primepi(79) = 22.
%H A181467 Artur Jasinski, <a href="/A181467/b181467.txt">Table of n, a(n) for n = 1..1698</a>
%t A181467 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 == 79, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t A181467 Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==79&]
%o A181467 (Magma) [ n: n in [2..300000] | m eq 79 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 21 2011
%o A181467 (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 79 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 21 2011
%o A181467 (PARI) is(n)=n=n^2-1; forprime(p=2, 73, n/=p^valuation(n, p)); n>1 && 79^valuation(n, 79)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A181467 Row 22 of A223701.
%Y A181467 Cf. A076605, A175607, A181447-A181466, A181468-A181470, A181568.
%K A181467 fini,full,nonn
%O A181467 1,1
%A A181467 _Artur Jasinski_, Oct 21 2010