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A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.

%I A128164 #128 Feb 16 2025 08:33:04
%S A128164 3,3,0,3,3,5,3,0,19,17,3,5,3,3,0,3,25667,19,3,3,5,5,3,0,7,3,5,5,5,7,0,
%T A128164 3,13,313,0,13,3,349,5,3,1319,5,5,19,7,127,19,0,3,4229,103,11,3,17,7,
%U A128164 3,41,3,7,7,3,5,0,19,3,19,5,3,29,3,7,5,5,3,41,3,3,5,3,0,23,5,17,5,11,7,61,3,3
%N A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.
%C A128164 a(n) = A084740(n) for all n except n = p-1, where p is an odd prime, for which A084740(n) = 2.
%C A128164 All nonzero terms are odd primes.
%C A128164 a(n) = 0 for n = {4,9,16,25,32,36,49,64,81,100,121,125,144,...}, which are the perfect powers with exceptions of the form n^(p^m) where p>2 and (n^(p^(m+1))-1)/(n^(p^m)-1) are prime and m>=1 (in which case a(n^(p^m))=p). - _Max Alekseyev_, Jan 24 2009
%C A128164 a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime.
%C A128164 a(152) > 20000. - _Eric Chen_, Jun 01 2015
%C A128164 a(n) is the least number k such that (n^k - 1)/(n-1) is a Brazilian prime, or 0 if no such Brazilian prime exists. - _Bernard Schott_, Apr 23 2017
%C A128164 These corresponding Brazilian primes are in A285642. - _Bernard Schott_, Aug 10 2017
%C A128164 a(152) = 270217, see the top PRP link. - _Eric Chen_, Jun 04 2018
%C A128164 a(184) = 16703, a(200) = 17807, a(210) = 19819, a(306) = 26407, a(311) = 36497, a(326) = 26713, a(331) = 25033; a(185) > 66337, a(269) > 63659, a(281) > 63421, and there are 48 unknown a(n) for n <= 1024. - _Eric Chen_, Jun 04 2018
%C A128164 Six more terms found: a(522)=20183, a(570)=12907, a(684)=22573, a(731)=15427, a(820)=12043, a(996)=14629. - _Michael Stocker_, Apr 09 2020
%H A128164 Max Alekseyev and Eric Chen, <a href="/A128164/b128164.txt">Table of n, a(n) for n = 2..184</a> (terms 2..151 from Max Alekseyev)
%H A128164 Eric Chen, <a href="/A128164/a128164_8.txt">Table of n, a(n) for n = 2..1024 status</a> (updated by Jinyuan Wang)
%H A128164 H. Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.
%H A128164 Richard Fischer, <a href="http://www.fermatquotient.com/PrimSerien/GenRepu.txt">Generalized repunit primes of the form (B^N-1)/(B-1)</a>
%H A128164 Top PRPs, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%28152%5En-1%29%2F%28152-1%29&amp;action=Search">Search by (152^n-1)/(152-1)</a>
%H A128164 Top PRPs, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-1%29%2Fa&amp;action=Search">Search by (b^n-1)/a</a>
%H A128164 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>
%e A128164 a(7) = 5 because (7^5 - 1)/6 = 2801 = 11111_7 is prime and (7^k - 1)/6 = 1, 8, 57, 400 for k = 1, 2, 3, 4. - _Bernard Schott_, Apr 23 2017
%t A128164 Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[(m^k - 1)/(m - 1)], k++]; k, 0]]@ If[Set[e, GCD @@ #[[All, -1]]] > 1, {#, IntegerExponent[n, #]} &@ Power[n, 1/e] /. {{k_, m_} /; Or[Not[PrimePowerQ@ m], Prime@ m, FactorInteger[m][[1, 1]] == 2] :> 0, {k_, m_} /; m > 1 :> n}, n] &@ FactorInteger@ n, {n, 2, 17}] (* _Michael De Vlieger_, Apr 24 2017 *)
%o A128164 (PARI) a052409(n) = my(k=ispower(n)); if(k, k, n>1)
%o A128164 a052410(n) = if (ispower(n, , &r), r, n)
%o A128164 is(n) = issquare(n) || (ispower(n) && !ispseudoprime((n^a052410(a052409(n))-1)/(n-1)))
%o A128164 a(n) = if(is(n), 0, forprime(p=3, 2^16, if(ispseudoprime((n^p-1)/(n-1)), return(p)))) \\ _Eric Chen_, Jun 01 2015, corrected by _Eric Chen_, Jun 04 2018, after _Charles R Greathouse IV_ in A052409 and _Michel Marcus_ in A052410
%Y A128164 Cf. A084738, A065854, A084740, A084741, A065507, A084742, A066180, A084732, A285642, A085104.
%Y A128164 Cf. A002384, A049409, A100330, A162862, A217070-A217089. (numbers b such that (b^p-1)/(b-1) is prime for prime p = 3 to 97)
%Y A128164 Cf. A000043, A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A133857, A006035, A127995, A127996, A127997, A204940, A127998, A127999, A128000, A181979, A098438, A128002, A209120, A185073, A128003, A128004, A181987, A128005, A239637, A240765, A294722, A242797, A243279, A267375, A245237, A245442, A173767. (numbers n such that (b^n-1)/(b-1) is prime for b = 2 to 53)
%Y A128164 A126589 gives locations of zeros.
%K A128164 nonn
%O A128164 2,1
%A A128164 _Alexander Adamchuk_, Feb 20 2007
%E A128164 a(18) = 25667 found by _Henri Lifchitz_, Sep 26 2007