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A208728 Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n. 46

%I #69 May 30 2019 08:58:29

%S 15,35,255,455,1295,2703,4355,6479,9215,10439,11951,16211,23435,27839,

%T 44099,47519,47879,62567,63167,65535,93023,94535,104195,120959,131327,

%U 133055,141155,142883,157079,170819,196811,207935,260831,283679,430199,560735,576719

%N Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n.

%C GCD(b,n)=1 and b^(n+1) == 1 (mod n).

%C The sequence lists the squarefree composite numbers n such that every prime divisor p of n satisfies (p-1)|(n+1) (similar to Korselt's criterion).

%C The sequence can be considered as an extension of k-Knödel numbers to k negative, in this case equal to -1.

%C Numbers n > 3 such that b^(n+2) == b (mod n) for every integer b. Also, numbers n > 3 such that A002322(n) divides n+1. Are there infinitely many such numbers? It seems that such numbers n > 35 have at least three prime factors. - _Thomas Ordowski_, Jun 25 2017

%H Charles R Greathouse IV, <a href="/A208728/b208728.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KorseltsCriterion.html">Korselt's Criterion</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnoedelNumbers.html">Knödel Numbers</a>

%e 6479 is part of the sequence because its prime factors are 11, 19 and 31: (6479+1)/(11-1)=648, (6479+1)/(19-1)=360 and (6479+1)/(31-1)=216.

%p with(numtheory); P:=proc(n) local d, ok, p;

%p if issqrfree(n) then p:=factorset(n); ok:=1;

%p for d from 1 to nops(p) do if frac((n+1)/(p[d]-1))>0 then ok:=0;

%p break; fi; od; if ok=1 then n; fi; fi; end: seq(P(i),i=5..576719);

%t Select[Range[2, 576719], SquareFreeQ[#] && ! PrimeQ[#] && Union[Mod[# + 1, Transpose[FactorInteger[#]][[1]] - 1]] == {0} &] (* _T. D. Noe_, Mar 05 2012 *)

%o (PARI) is(n)=if(isprime(n)||!issquarefree(n)||n<3, return(0)); my(f=factor(n)[, 1]); for(i=1, #f, if((n+1)%(f[i]-1), return(0))); 1 \\ _Charles R Greathouse IV_, Mar 05 2012

%Y Cf. A002322, A002997, A006972, A033553, A050990, A050992, A050993, A208154-A208158.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Mar 01 2012

%E Definition corrected by _Thomas Ordowski_, Jun 25 2017

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Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)