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

1,1

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

Prog

(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

Crossrefs

Cf. A104016, A002997.

Sequence in context: A252859 A177216 A112441 * A317400 A284814 A228627

Adjacent sequences: A104014 A104015 A104016 * A104018 A104019 A104020

Keyword

nonn

Author

Max Alekseyev, Feb 25 2005

Status

approved