

A216911


Prime factors of Carmichael numbers divisible by 3, taken just once each as it appears first time, in order of the size of the Carmichael number respectively in order of their size if they are prime factors of the same Carmichael number.


1



3, 11, 17, 5, 47, 89, 101, 197, 29, 263, 521, 1559, 173, 3011, 71, 641, 1277, 53, 317, 4583, 617, 4019, 401, 3041, 41, 479, 3347, 131, 10427, 4643, 1301, 419, 6689, 5531, 281, 55217, 251, 2417, 4001, 491, 1601, 3137, 449, 3617, 107, 2969, 4211, 6737, 1061
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OFFSET

1,1


COMMENTS

Every number from sequence, beside 3, is a prime of the form 6*k  1.
It is remarkable that, if we note with p1 and p2 two numbers from sequence, beside 3, through formula p1*p2  3*p1  p2 + 4 are obtained extremely many products of the form 3^n*d^m, where d is prime or the unit.
For instance, taken p2 = 5 and p1 other prime from sequence, beside 3, were obtained through the formula 2*p1  1 the following primes d (in the brackets is the prime p1): 7(11), 11(17), 31(47), 59(89), 67(101), 131(197), 19(29), 347(521), 1039(1559), 223(3011), 47(71), 211(317), 137(617), 89(401), 2027(3041), 1(41), 29(131), 17(1301), 31(619), 347(521), 1229(5531), 167(251), 179(2417), 109(491), 2411(3617), 71(107), 1979(2969), 499(6737). For the numbers from sequence of the form 10*h + 3 were obtained through this formula the following primes d multiplied, this time, with 3 and 5 or powers of 3 and 5: 7(263), 23(173), 7(53), 619(4643). For the numbers from sequence for which is not obtained through this formula a prime or a power of prime multiplied with 3 and 5 or powers of 3 and 5 is obtained a product of two primes, not necessary squarefree.
The formula p1*p2  3*p1  p2 + 4 generates many primes also if is applied to the prime factors of the same Carmichael number (not necessary divisible by 3); taken, for instance, 1729 = 7*13*19 were obtained the following primes or powers of primes (in the brackets are p1 and p2): 61(7,13), 7^2(13,7), 97(7,19), 73(19,7), 193(13,19), 181(19,13).


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..3744
Eric Weisstein's World of Mathematics, Carmichael Number


CROSSREFS

Cf. A002997.
Sequence in context: A029500 A243770 A298701 * A154497 A322171 A038946
Adjacent sequences: A216908 A216909 A216910 * A216912 A216913 A216914


KEYWORD

nonn


AUTHOR

Marius Coman, Sep 20 2012


EXTENSIONS

a(26), a(34), a(43) corrected by Charles R Greathouse IV, Sep 20 2012


STATUS

approved



