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A212882 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. 1
63973, 172081, 31146661, 167979421, 277241401, 703995733, 1504651681, 1949646601, 2414829781, 21595159873, 117765525241, 192739365541, 461574735553, 881936608681, 2732745608209, 3145699746793, 3307287048121, 3976486324993, 7066238244481, 7932245192461, 8916642713161, 9924090391909 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

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

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

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

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

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

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

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

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

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

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).

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).

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

E. W. Weisstein, Carmichael Number

E. W. Weisstein, Korselt’s Criterion

PROG

(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

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

CROSSREFS

Sequence in context: A236608 A214758 A265827 * A290793 A182518 A317136

Adjacent sequences:  A212879 A212880 A212881 * A212883 A212884 A212885

KEYWORD

nonn

AUTHOR

Marius Coman, May 29 2012

EXTENSIONS

a(8) and a(10) inserted by Charles R Greathouse IV, Jul 07 2017

STATUS

approved

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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)