A182207 Carmichael numbers of the form C = p*(2p-1)*(n*(2p-2)+p), where p and 2p-1 are prime numbers.

1729, 2821, 41041, 63973, 101101, 126217, 172081, 188461, 294409, 399001, 488881, 512461, 670033, 748657, 838201, 852841, 997633, 1033669, 1050985, 1082809, 1461241, 2100901, 2113921, 2628073, 4463641, 4909177, 7995169, 8341201, 8719309, 9890881

Offset

1,1

Comments

Conjecture: Any Carmichael number C divisible by p and 2p-1 (where p and 2p-1 are prime numbers) can be written as C = p*(2p-1)*(n*(2p-2)+p).

Checked for the first 30 Carmichael numbers divisible by p and 2p-1.

Note: we can see how easy is to obtain Carmichael numbers with this formula:

for n = 1 we get p*(2p-1)*(3p-2) and Carmichael numbers 1729, 172081, 294409 etc.

for n = 2 we get p*(2p-1)*(5p-4) and Carmichael numbers 2821, 63973, 488881 etc.

for n = 3 we get p*(2p-1)*(7p-6) and Carmichael numbers 399001, 53711113 etc.

Links

Table of n, a(n) for n=1..30.

E. W. Weisstein, Carmichael Number

Prog

(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
list(lim)=my(v=List(), q, t); forprime(p=3, round(solve(x=1, lim, 6*x^3-7*x^2+2*x-lim)), for(n=1, (lim\(2*p^2-p)-p)\(2*p-2), if(isprime(q=2*p-1)&&Korselt(t=p*q*(n*q-n+p)), listput(v, t)))); vecsort(Vec(v), , 8)
\\ Charles R Greathouse IV, Oct 02 2012

Crossrefs

Sequence in context: A182208 A340092 A324316 * A138129 A242880 A001235

Adjacent sequences: A182204 A182205 A182206 * A182208 A182209 A182210

Keyword

nonn

Author

Marius Coman, Apr 18 2012

Status

approved