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
KEYWORD
nonn
AUTHOR
Max Alekseyev, Feb 25 2005
STATUS
approved