A181454 Numbers n such that 23 is the largest prime factor of n^2 - 1.

22, 24, 45, 47, 91, 116, 137, 139, 183, 208, 229, 254, 298, 321, 323, 344, 415, 461, 505, 551, 599, 645, 781, 783, 919, 967, 1013, 1057, 1126, 1151, 1310, 1471, 1519, 1749, 1793, 2186, 2209, 2276, 2393, 2575, 2874, 2991, 3704, 3725, 4047, 4049, 4369

Offset

1,1

Comments

Sequence is finite, for proof see A175607.

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

Links

A. Jasinski, Table of n, a(n) for n = 1..95 (full sequence)

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

Prog

(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
(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
(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

Crossrefs

Cf. A175607, A181447-A181453, A181455-A181470, A181568.

Sequence in context: A235807 A244397 A138603 * A155911 A061411 A053779

Adjacent sequences: A181451 A181452 A181453 * A181455 A181456 A181457

Keyword

fini,full,nonn

Author

Artur Jasinski, Oct 21 2010

Status

approved