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A181470
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Numbers n such that 97 is the largest prime factor of n^2 - 1.
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27
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96, 98, 193, 195, 290, 389, 484, 581, 583, 775, 872, 874, 969, 971, 1066, 1163, 1165, 1359, 1456, 1551, 1553, 1648, 1747, 1844, 1939, 2036, 2133, 2135, 2232, 2521, 2715, 2911, 3008, 3103, 3299, 3394, 3396, 3590, 3976, 4267, 4269, 4463, 4558, 4946, 5045
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OFFSET
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1,1
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COMMENTS
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Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(25) = 99913980938200001; primepi(97) = 25.
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LINKS
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Lucas A. Brown, Table of n, a(n) for n = 1..2734 (2679 terms from Artur Jasinski, 55 missing terms from Lucas A. Brown, complete subject to the effective abc conjecture c < rad(abc)^2)
To verify full I listed all 97-smooth numbers k that are a multiple of 97 below (inclusive) A175607(25) + 2. I then checked if k+2 is 97-smooth. If so, k+1 is a term. Then similarily I checked if k-2 is 97-smooth. If so, k-1 is a term.
Doing so found the 2734 terms from the b-file. All candidates have been checked completing the proof of full. (End)
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MATHEMATICA
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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 == 97, AppendTo[rr, n]]]; n++]; rr
(* or *)
Select[Range[300000], FactorInteger[#^2 - 1][[-1, 1]] == 97 &]
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PROG
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(Magma) [ n: n in [2..300000] | m eq 97 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 97 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 2011
(PARI) is(n)=n=n^2-1; forprime(p=2, 89, n/=p^valuation(n, p)); n>1 && 97^valuation(n, 97)==n \\ Charles R Greathouse IV, Jul 01 2013
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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