A181454 - OEIS

%I #26 Dec 20 2024 18:15:58

%S 22,24,45,47,91,116,137,139,183,208,229,254,298,321,323,344,415,461,

%T 505,551,599,645,781,783,919,967,1013,1057,1126,1151,1310,1471,1519,

%U 1749,1793,2186,2209,2276,2393,2575,2874,2991,3704,3725,4047,4049,4369

%N Numbers k such that 23 is the largest prime factor of k^2 - 1.

%C Numbers k such that A076605(k) = 23.

%C Sequence is finite, for proof see A175607.

%C Search for terms can be restricted to the range from 2 to A175607(9) = 10285001; primepi(23) = 9.

%H Artur Jasinski, <a href="/A181454/b181454.txt">Table of n, a(n) for n = 1..95</a> (full sequence)

%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 < 14000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 23, AppendTo[rr, n]]]; n++ ]; rr

%t Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==23&]

%o (Magma) [ n: n in [2..300000] | m eq 23 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // _Klaus Brockhaus_, Feb 18 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..10300000] | p mod (n^2-1) eq 0 and (D[#D] eq 23 where D is PrimeDivisors(n^2-1)) ]; // _Klaus Brockhaus_, Feb 24 2011

%o (PARI) is(n)=n=n^2-1; forprime(p=2, 19, n/=p^valuation(n, p)); n>1 && 23^valuation(n, 23)==n \\ _Charles R Greathouse IV_, Jul 01 2013

%Y Cf. A076605, A175607, A181447-A181453, A181455-A181470, A181568.

%K fini,full,nonn

%O 1,1

%A _Artur Jasinski_, Oct 21 2010