A182085 - OEIS

%I #33 Mar 06 2023 11:19:22

%S 2821,488881,288120421,492559141,776176261,1632785701,3835537861,

%T 6735266161,9030158341,21796387201,167098039921,288374745541,

%U 351768558961,381558955141,505121232001,582561482161,915245066821,2199733160881,2402435763841,4541477778181

%N Carmichael numbers of the form (30k+7)*(60k+13)*(150k+31).

%C Note that in this sequence, 30k+7, 60k+13, and 150k+31 do not have to be prime. These numbers were found by taking the intersection of Carmichael numbers found by Pinch and numbers of the form (30k+7)*(60k+13)*(150k+31).

%C Conjecture: N = (30k+7)*(60k+13)*(150k+31) is a Carmichael number if (but not only if) 30k+7, 60k+13 and 150k+31 are all three prime numbers.

%C We checked the conjecture up to k = 256; we got Carmichael numbers with three prime divisors for k = 0, 1, 10, 12, 18, 24, 32, 43, 85, 102, 123, 129, 150, 201, 207, 256.

%C We got Carmichael numbers with more than three prime divisors for n = 14, 29, 109, 112.

%C All these numbers can be written as well as N = (n+1)*(2n+1)*(5n+1), where n = 30k+6.

%C The conjecture follows from Korselt's criterion. - _Charles R Greathouse IV_, Oct 02 2012

%H Charles R Greathouse IV, <a href="/A182085/b182085.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>.

%o (PARI) test(lim)={

%o     my(v=List(),n,f);

%o     for(k=0,lim,

%o         n=(30*k+7)*(60*k+13)*(150*k+31)-1;

%o         f=factor(30*k+7);

%o         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));

%o         f=factor(60*k+13);

%o         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));

%o         f=factor(150*k+31);

%o         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));

%o         listput(v,n+1)

%o     );

%o     Vec(v)

%o }; \\ _Charles R Greathouse IV_, Oct 02 2012

%Y Cf. A002997 (Carmichael numbers), A087788.

%K nonn

%O 1,1

%A _Marius Coman_, Apr 11 2012

%E Extended and corrected by _T. D. Noe_, Apr 19 2012