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Search: id:A104016
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| A104016 |
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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). |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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A. K. Devaraj conjectured that these numbers are exactly Carmichael numbers. It was proved http://www.mersenneforum.org/showpost.php?p=55271 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. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 03 2009]
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PROGRAM
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(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)) ) }
Contribution from M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 03 2009: (Start)
(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 \\ (End)
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CROSSREFS
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Cf. A104017, A002997.
Sequence in context: A135721 A047713 A006971 this_sequence A002997 A087788 A083733
Adjacent sequences: A104013 A104014 A104015 this_sequence A104017 A104018 A104019
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KEYWORD
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nonn
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AUTHOR
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Max Alekseyev (maxale(AT)gmail.com), Feb 25 2005
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