A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.

42, 44, 85, 87, 171, 173, 216, 257, 259, 300, 343, 386, 431, 474, 517, 560, 601, 687, 689, 730, 818, 859, 1074, 1117, 1119, 1289, 1291, 1332, 1420, 1549, 1633, 1721, 1805, 1891, 1977, 1979, 2108, 2321, 2495, 2665, 2667, 2751, 2753, 2794, 2925, 3095, 3484

Offset: 1

Author

Artur Jasinski, Oct 21 2010

Keywords

• fini
• full
• nonn

Comments

Numbers k such that A076605(k) = 43.

Sequence is finite, for proof see A175607.

Search for terms can be restricted to the range from 2 to A175607(14) = 842277599279; primepi(43) = 14.

Links

• Artur Jasinski, Table of n, a(n) for n = 1..343

Crossrefs

Cf. A076605, A175607, A181447-A181458, A181460-A181470, A181568.

Programs

Magma

[ n: n in [2..300000] | m eq 43 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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 43 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

Mathematica

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 == 43, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==43&]

PARI

is(n)=n=n^2-1; forprime(p=2, 41, n/=p^valuation(n, p)); n>1 && 43^valuation(n, 43)==n \\ Charles R Greathouse IV, Jul 01 2013