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

30, 32, 61, 63, 92, 94, 125, 154, 185, 249, 309, 311, 342, 373, 404, 433, 495, 526, 528, 559, 681, 683, 714, 869, 898, 929, 991, 1055, 1084, 1177, 1241, 1301, 1427, 1520, 1611, 1673, 1735, 1799, 1861, 1921, 1954, 2047, 2107, 2419, 2696, 2729, 2851, 3037

Offset: 1

Author

Artur Jasinski, Oct 21 2010

Keywords

• fini
• full
• nonn

Comments

Numbers k such that A076605(k) = 31.

Sequence is finite, for proof see A175607.

Search for terms can be restricted to the range from 2 to A175607(11) = 3222617399; primepi(31) = 11.

Links

• Artur Jasinski, Table of n, a(n) for n = 1..168 (full sequence)

Crossrefs

Cf. A076605, A175607, A181447-A181455, A181457-A181470, A181568.

Programs

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

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

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 == 31, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[5000],Max[Transpose[FactorInteger[ #^2-1]][[1]]]==31&] (* Harvey P. Dale, Nov 03 2010 *)

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