A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1*...*pr such that phi(N)=(p1-1)*...*(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 11305, 15841, 29341, 39865, 41041, 46657, 52633, 62745, 63973, 75361, 96985, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 401401, 410041

Offset: 1

Author

Max Alekseyev, Feb 25 2005

Keywords

• nonn

Comments

A.K. Devaraj conjectured that these numbers are exactly Carmichael numbers. It was proved (see Alekseyev link) that every Carmichael number is indeed a Devaraj number, but the converse is not true. Devaraj numbers that are not Carmichael are given by A104017.

These numbers can't be even, since phi(N) is always even (N>2) but p1=2 implies that gcd{pi-1}=1 and N-1 is odd. - M. F. Hasler, Apr 03 2009

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
• Max Alekseyev, Pomerance's proof, June 2005.

Crossrefs

Subsequence of A350352 and hence of A033942.

Cf. A104017, A002997.

Programs

PARI

Devaraj() = for(n=2,10^8, f=factorint(n); if(vecmax(f[,2])>1,next); f=f[,1]; r=length(f); if(r==1,next); d=f[1]-1; p=f[1]-1; for(i=2,r,d=gcd(d,f[i]-1); p*=f[i]-1); if( ((n-1)^(r-2)*d^2)%p==0, print1(" ",n)) )

PARI

isA104016(n)= local(f=factor(n)); vecmax(f[,2])==1 && #(f*=[1,-1]~)>1 && gcd(f)^2*(n-1)^(#f-2)%prod(i=1,#f,f[i])==0
/* To print the list: */ forstep( n=3, 10^6, 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]) || print1(n","))
/* The following version could be efficient for large omega(n) */
isA104016(n) = issquarefree(n) && !isprime(n) && Mod(n-1,prod(i=1,#n=factor(n)*[1,-1]~,n[i]))^(#n-2)*gcd(n)^2==0 \\ M. F. Hasler, Apr 03 2009