A181461 Numbers k such that 53 is the largest prime factor of k^2-1.

52, 54, 105, 107, 160, 211, 319, 370, 476, 529, 531, 584, 637, 741, 743, 847, 849, 900, 902, 953, 1059, 1220, 1273, 1324, 1377, 1379, 1483, 1538, 1644, 1695, 1801, 1803, 2015, 2174, 2278, 2386, 2437, 2543, 2651, 2755, 2861, 2969, 3073, 3181, 3497, 3499

Offset: 1

Author

Artur Jasinski, Oct 21 2010

Keywords

• fini
• full
• nonn

Comments

Numbers k such that A076605(k) = 53.

Sequence is finite, for proof see A175607.

Search for terms can be restricted to the range from 2 to A175607(16) = 2907159732049; primepi(53) = 16.

Links

• Artur Jasinski, Table of n, a(n) for n = 1..507

Crossrefs

Cf. A076605, A175607, A181447-A181460, A181462-A181470, A181568.

Programs

Magma

[ n: n in [2..300000] | m eq 53 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 53 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 == 53, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==53&]

PARI

is(n)=n=n^2-1; forprime(p=2, 47, n/=p^valuation(n, p)); n>1 && 53^valuation(n, 53)==n \\ Charles R Greathouse IV, Jul 01 2013