A181457 - OEIS

%I #23 Dec 20 2024 18:16:31

%S 36,38,73,75,149,186,221,223,260,295,369,371,406,443,482,519,593,628,

%T 776,813,815,961,1000,1072,1259,1331,1333,1405,1407,1444,1481,1701,

%U 1814,1849,1886,1923,1999,2071,2367,2591,2663,2737,2887,2959,3329,3331,3403

%N Numbers k such that 37 is the largest prime factor of k^2 - 1.

%C Numbers k such that A076605(k) = 37.

%C Sequence is finite, for proof see A175607.

%C Search for terms can be restricted to the range from 2 to A175607(12) = 9447152318; primepi(37) = 12.

%H Artur Jasinski, <a href="/A181457/b181457.txt">Table of n, a(n) for n = 1..208</a>

%t 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 == 37, AppendTo[rr, n]]]; n++ ]; rr (* _Artur Jasinski_ *)

%t Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==37&]

%o (Magma) [ n: n in [2..300000] | m eq 37 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 19 2011

%o (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 37 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 20 2011

%o (PARI) is(n)=n=n^2-1; forprime(p=2, 31, n/=p^valuation(n, p)); n>1 && 37^valuation(n, 37)==n \\ _Charles R Greathouse IV_, Jul 01 2013

%Y Cf. A076605, A175607, A181447-A181456, A181458-A181470, A181568.

%K fini,full,nonn

%O 1,1

%A _Artur Jasinski_, Oct 21 2010