A276818 Composite numbers n such that b^gpf(n) == b (mod n) for every integer b, where gpf(n) = A006530(n).

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 38, 39, 42, 46, 51, 57, 58, 62, 65, 66, 69, 74, 78, 82, 85, 86, 87, 91, 93, 94, 102, 106, 111, 114, 118, 122, 123, 129, 130, 133, 134, 138, 141, 142, 145, 146, 158, 159, 166, 170, 174, 177, 178, 182, 183, 185, 186, 194

Offset

1,1

Comments

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

Squarefree composite numbers n such that for every prime p dividing n, p-1 divides gpf(n)-1. - Robert Israel, Sep 18 2016

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

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

Squarefree composite numbers n such that A002322(n) = gpf(n)-1. - Thomas Ordowski, Feb 25 2018

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

filter:= proc(n) local F, p;
  if isprime(n) or not numtheory:-issqrfree(n) then return false fi;
  F:= numtheory:-factorset(n);
  p:= max(F);
  evalb(map(t -> (p-1) mod (t-1), F) = {0})
end proc:
select(filter, [$2..1000]); # Robert Israel, Sep 18 2016

Mathematica

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A120944 A327829 A052053 * A325259 A320911 A371781

Adjacent sequences: A276815 A276816 A276817 * A276819 A276820 A276821

Keyword

nonn

Author

Thomas Ordowski, Sep 18 2016

Extensions

More terms from Michel Marcus and Robert Israel, Sep 18 2016

Status

approved