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%I #26 Mar 13 2023 08:24:55
%S 11305,39865,96985,401401,464185,786961,1106785,1296505,1719601,
%T 1993537,2242513,2615977,2649361,2722681,3165961,3181465,3755521,
%U 4168801,4229601,4483297,4698001,5034601,5381265,5910121,5977153,7177105
%N Devaraj numbers (A104016) which are not Carmichael numbers.
%C Counterexamples to Devaraj's 2nd conjecture: _A.K. Devaraj_ conjectured that these numbers are exactly Carmichael numbers. It was proved (see A104016 ) that every Carmichael number is indeed a Devaraj number, but the converse is not true. Devaraj numbers that are not Carmichael are listed here.
%C It is sufficient to scan only odd numbers (cf. A104016), which makes the computation of the list twice as fast. - _M. F. Hasler_, Apr 03 2009
%H Charles R Greathouse IV, <a href="/A104017/b104017.txt">Table of n, a(n) for n = 1..10000</a>
%o (PARI) DNC() = for(n=2,10^8, f=factorint(n); if(vecmax(f[,2])>1,next); f=f[,1]; r=length(f); if(r==1,next); Carmichael=1; d=f[1]-1; p=1; for(i=1,r, d=gcd(d,f[i]-1); p*=f[i]-1; if((n-1)%(f[i]-1),Carmichael=0)); if( ((n-1)^(r-2)*d^2)%p==0 && !Carmichael, print1(" ",n)) )
%o (PARI) forstep( n=3, 10^7, 2, vecmax((f=factor(n))[,2])>1 && next; #(f*=[1,-1]~)>1 || next; gcd(f)^2*(n-1)^(#f-2) % prod(i=1,#f,f[i]) && next; for( i=1,#f, (n-1)%f[i] && !print1(n",") && break)) \\ _M. F. Hasler_, Apr 03 2009
%o (PARI) Korselt(n,p)=for(i=1, #p, if((n-1)%(p[i]-1), return(0))); 1
%o Devaraj(n,p)=my(u=apply(q->q-1,p)); gcd(u)^2*(n-1)^(#p-2)%vecprod(u)==0
%o list(lim)=my(v=List()); forsquarefree(N=11305,lim\=1, my(p=N[2][,1],n=N[1]); if(p[1]>2 && #p>2 && Devaraj(n,p) && !Korselt(n,p), listput(v,n))); Vec(v) \\ _Charles R Greathouse IV_, Mar 09 2023
%Y Cf. A104016, A002997.
%K nonn
%O 1,1
%A _Max Alekseyev_, Feb 25 2005
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