A212882 - OEIS

%I #23 Jul 10 2017 09:14:25

%S 63973,172081,31146661,167979421,277241401,703995733,1504651681,

%T 1949646601,2414829781,21595159873,117765525241,192739365541,

%U 461574735553,881936608681,2732745608209,3145699746793,3307287048121,3976486324993,7066238244481,7932245192461,8916642713161,9924090391909

%N Carmichael numbers of the form n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1), where p is odd, p from 3 to 23.

%C The following Carmichael numbers are of the form n*(2n-1)*(3n-2)*(6n-5): 63973, 31146661, 703995733, 2414829781, 192739365541, 461574735553, 3976486324993.

%C The following Carmichael numbers are of the form n*(2n-1)*(5n-4)*(10n-9): 172081, 881936608681, 3307287048121, 8916642713161.

%C The following Carmichael number is of the form n*(2n-1)*(7n-6)*(14n-13): 167979421.

%C The following Carmichael number is of the form n*(2n-1)*(9n-8)*(18n-17): 277241401.

%C The following Carmichael number is of the form n*(2n-1)*(11n-10)*(22n-21): 9924090391909.

%C The following Carmichael number is of the form n*(2n-1)*(15n-14)*(30n-29): 7932245192461.

%C The following Carmichael number is of the form n*(2n-1)*(17n-16)*(34n-33): 3145699746793.

%C The following Carmichael numbers are of the form n*(2n-1)*(21n-20)*(42n-41): 1504651681, 117765525241, 2732745608209.

%C The following Carmichael number is of the form n*(2n-1)*(23n-22)*(46n-45): 7066238244481.

%C For p=13 and p=19, there is no Carmichael number up to 10^13.

%C There is not any other Carmichael number of this form, for p from 3 to 23, up to 10^13.

%C Conjecture: for any odd number p we have an infinite number of Carmichael numbers of the form n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1).

%C Note: many numbers of the form n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1), not divisible by 2, 3 or 5, where p is odd or even, are squarefree and respects the Korselt's criterion for many of their prime divisors or are not squarefree but respects the Korselt's criterion sometimes even for all their divisors (but we didn’t find Carmichael numbers when p is even).

%H Charles R Greathouse IV, <a href="/A212882/b212882.txt">Table of n, a(n) for n = 1..10000</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/KorseltsCriterion.html">Korselt’s Criterion</a>

%o (PARI) Kv(n, v)=for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, my(f=factor(v[i])); for(j=1, #f~, if(f[j, 2]>1 || (n-1)%(f[j, 1]-1), return(0)))); 1

%o list(lim)=my(v=List(),n,C); forstep(p=3,23,2, n=3; while((C=n*(2*n - 1)*(p*n - p + 1)*(2*p*n - 2*p + 1))<=lim, if(Kv(C,[n, 2*n-1, p*n-p+1, 2*p*n-2*p+1]), listput(v,C)); n+=2)); Set(v) \\ _Charles R Greathouse IV_, Jul 07 2017

%K nonn

%O 1,1

%A _Marius Coman_, May 29 2012

%E a(8) and a(10) inserted by _Charles R Greathouse IV_, Jul 07 2017