A181466 Numbers k such that 73 is the largest prime factor of k^2 - 1.

72, 74, 145, 147, 218, 220, 291, 293, 364, 439, 512, 729, 731, 804, 875, 1021, 1023, 1167, 1169, 1240, 1313, 1315, 1459, 1461, 1607, 1678, 1680, 1751, 1826, 1899, 2045, 2116, 2262, 2481, 2483, 2554, 2702, 2773, 2848, 3067, 3284, 3359, 3576, 3649, 3722

Offset: 1

Author

Artur Jasinski, Oct 21 2010

Keywords

• fini
• full
• nonn

Comments

Numbers k such that A076605(k) = 73.

Sequence is finite, for proof see A175607.

Search for terms can be restricted to the range from 2 to A175607(21) = 4573663454608289; primepi(73) = 21.

Links

• Artur Jasinski, Table of n, a(n) for n = 1..1439

Crossrefs

Row 21 of A223701

Cf. A076605, A175607, A181447-A181465, A181467-A181470, A181568.

Programs

Magma

[ n: n in [2..300000] | m eq 73 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 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 73 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 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 == 73, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==73&]

PARI

is(n)=n=n^2-1; forprime(p=2, 71, n/=p^valuation(n, p)); n>1 && 73^valuation(n, 73)==n \\ Charles R Greathouse IV, Jul 01 2013