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A181456 Numbers k such that 31 is the largest prime factor of k^2 - 1.

%I A181456 #34 Dec 20 2024 18:16:22
%S A181456 30,32,61,63,92,94,125,154,185,249,309,311,342,373,404,433,495,526,
%T A181456 528,559,681,683,714,869,898,929,991,1055,1084,1177,1241,1301,1427,
%U A181456 1520,1611,1673,1735,1799,1861,1921,1954,2047,2107,2419,2696,2729,2851,3037
%N A181456 Numbers k such that 31 is the largest prime factor of k^2 - 1.
%C A181456 Numbers k such that A076605(k) = 31.
%C A181456 Sequence is finite, for proof see A175607.
%C A181456 Search for terms can be restricted to the range from 2 to A175607(11) = 3222617399; primepi(31) = 11.
%H A181456 Artur Jasinski, <a href="/A181456/b181456.txt">Table of n, a(n) for n = 1..168</a> (full sequence)
%t A181456 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 == 31, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)
%t A181456 Select[Range[5000],Max[Transpose[FactorInteger[ #^2-1]][[1]]]==31&] (* _Harvey P. Dale_, Nov 03 2010 *)
%o A181456 (Magma) [ n: n in [2..300000] | m eq 31 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 19 2011
%o A181456 (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 31 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 20 2011
%o A181456 (PARI) is(n)=n=n^2-1; forprime(p=2, 29, n/=p^valuation(n, p)); n>1 && 31^valuation(n, 31)==n \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A181456 Cf. A076605, A175607, A181447-A181455, A181457-A181470, A181568.
%K A181456 fini,full,nonn
%O A181456 1,1
%A A181456 _Artur Jasinski_, Oct 21 2010