A276818 - OEIS

%I #60 Mar 10 2018 10:50:32

%S 6,10,14,15,21,22,26,30,33,34,38,39,42,46,51,57,58,62,65,66,69,74,78,

%T 82,85,86,87,91,93,94,102,106,111,114,118,122,123,129,130,133,134,138,

%U 141,142,145,146,158,159,166,170,174,177,178,182,183,185,186,194

%N Composite numbers n such that b^gpf(n) == b (mod n) for every integer b, where gpf(n) = A006530(n).

%C It suffices to check all bases 2 <= b <= (n+1)/2.

%C Squarefree composite numbers n such that for every prime p dividing n, p-1 divides gpf(n)-1. - _Robert Israel_, Sep 18 2016

%C An even number 2m is in the sequence iff m is an odd prime or odd m is in the sequence. - _Altug Alkan_ and _Thomas Ordowski_, Sep 19 2016

%C Problem: are there infinitely many Carmichael numbers A002997 in the sequence? These are Carmichael numbers m such that m/gpf(m) is a Carmichael number, they are a proper subset of A214758. - _Thomas Ordowski_ and _Altug Alkan_, Sep 19 2016

%C Squarefree composite numbers n such that A002322(n) = gpf(n)-1. - _Thomas Ordowski_, Feb 25 2018

%H Robert Israel, <a href="/A276818/b276818.txt">Table of n, a(n) for n = 1..10000</a>

%p filter:= proc(n) local F,p;

%p   if isprime(n) or not numtheory:-issqrfree(n) then return false fi;

%p   F:= numtheory:-factorset(n);

%p   p:= max(F);

%p   evalb(map(t -> (p-1) mod (t-1), F) = {0})

%p end proc:

%p select(filter, [$2..1000]); # _Robert Israel_, Sep 18 2016

%t Select[DeleteCases[Range@ 200, k_ /; ! CompositeQ@ k], Function[n, Times @@ Boole@ Map[Mod[#, n] == Mod[#^(FactorInteger[n][[-1, 1]]), n] &, Range[2, Floor[(n + 1)/2]]] == 1]] (* _Michael De Vlieger_, Sep 19 2016 *)

%o (PARI) is(n)=if(n%2==0, if(n%4, if(isprime(n/2), return(1), n>>=1), return(0))); my(f=factor(n)); if(#f~ < 2 || vecmax(f[,2])>1, return(0)); for(i=1,#f~, if((f[#f~,1]-1)%(f[i,1]-1), return(0))); 1 \\ _Charles R Greathouse IV_, Sep 19 2016

%o (PARI) lista(nn) = forcomposite(n=1, nn, if ( issquarefree(n) && !((vecmax(factor(n)[,1]) - 1) % lcm(znstar(n)[2])), print1(n, ", "))); \\ _Michel Marcus_, Sep 11 2017

%Y Cf. A002322, A002997, A006530, A177516, A276832.

%K nonn

%O 1,1

%A _Thomas Ordowski_, Sep 18 2016

%E More terms from _Michel Marcus_ and _Robert Israel_, Sep 18 2016