

A128164


Least k > 2 such that (n^k  1)/(n1) is prime, or 0 if no such prime exists.


9



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, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 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
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OFFSET

2,1


COMMENTS

a(n) = A084740(n) for all n except n = p1, where p is an odd prime, for which A084740(n) = 2.
All nonzero terms are odd primes.
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
a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime.
a(n) is the least number k such that (n^k  1)/(n1) is a Brazilian prime, or 0 if no such Brazilian prime exists.  Bernard Schott, Apr 23 2017
a(152) = 270217, see the top PRP link.  Eric Chen, Jun 04 2018
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
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


LINKS

Eric Weisstein's World of Mathematics, Repunit


EXAMPLE

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


MATHEMATICA

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 *)


PROG

(PARI) a052409(n) = my(k=ispower(n)); if(k, k, n>1)
a052410(n) = if (ispower(n, , &r), r, n)
is(n) = issquare(n)  (ispower(n) && !ispseudoprime((n^a052410(a052409(n))1)/(n1)))


CROSSREFS

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^n1)/(b1) is prime for b = 2 to 53)


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



