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%I #127 Feb 10 2024 03:00:20
%S 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 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 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 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.
%C a(n) = A084740(n) for all n except n = p-1, where p is an odd prime, for which A084740(n) = 2.
%C All nonzero terms are odd primes.
%C 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 a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime.
%C a(152) > 20000. - _Eric Chen_, Jun 01 2015
%C 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 These corresponding Brazilian primes are in A285642. - _Bernard Schott_, Aug 10 2017
%C a(152) = 270217, see the top PRP link. - _Eric Chen_, Jun 04 2018
%C 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 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 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 Eric Chen, <a href="/A128164/a128164_8.txt">Table of n, a(n) for n = 2..1024 status</a> (updated by Jinyuan Wang)
%H 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 Richard Fischer, <a href="http://www.fermatquotient.com/PrimSerien/GenRepu.txt">Generalized repunit primes of the form (B^N-1)/(B-1)</a>
%H Top PRPs, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%28152%5En-1%29%2F%28152-1%29&action=Search">Search by (152^n-1)/(152-1)</a>
%H Top PRPs, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-1%29%2Fa&action=Search">Search by (b^n-1)/a</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%e 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 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 (PARI) a052409(n) = my(k=ispower(n)); if(k, k, n>1)
%o a052410(n) = if (ispower(n, , &r), r, n)
%o is(n) = issquare(n) || (ispower(n) && !ispseudoprime((n^a052410(a052409(n))-1)/(n-1)))
%o 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 Cf. A084738, A065854, A084740, A084741, A065507, A084742, A066180, A084732, A285642, A085104.
%Y 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 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 A126589 gives locations of zeros.
%K nonn
%O 2,1
%A _Alexander Adamchuk_, Feb 20 2007
%E a(18) = 25667 found by _Henri Lifchitz_, Sep 26 2007
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