A104017 Devaraj numbers (A104016) which are not Carmichael numbers.

11305, 39865, 96985, 401401, 464185, 786961, 1106785, 1296505, 1719601, 1993537, 2242513, 2615977, 2649361, 2722681, 3165961, 3181465, 3755521, 4168801, 4229601, 4483297, 4698001, 5034601, 5381265, 5910121, 5977153, 7177105

Offset: 1

Author

Max Alekseyev, Feb 25 2005

Keywords

• nonn

Comments

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.

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A104016, A002997.

Programs

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)) )

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

PARI

Korselt(n,p)=for(i=1, #p, if((n-1)%(p[i]-1), return(0))); 1
Devaraj(n,p)=my(u=apply(q->q-1,p)); gcd(u)^2*(n-1)^(#p-2)%vecprod(u)==0
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