A216830 Prime factors of Carmichael numbers divisible by 7, 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.

7, 13, 19, 31, 23, 41, 67, 73, 11, 103, 37, 101, 61, 109, 199, 433, 5, 17, 151, 577, 307, 163, 139, 181, 271, 739, 229, 251, 853, 1321, 991, 241, 53, 397, 1783, 1171, 907, 2971, 353, 593, 4057, 661, 193, 619, 89, 653, 157, 2089, 313, 331, 373, 937, 2053, 443, 3877

Offset: 1

Author

Marius Coman, Sep 19 2012

Keywords

• nonn

Comments

It is remarkable that, if we note with p the numbers from sequence, for every p was obtained a prime, a squarefree semiprime or a number divisible by 5 through the formula 3*p + 4.

Primes obtained and the corresponding p in the brackets: 43(13), 61(19), 97(31), 73(23), 127(41), 223(73), 37(11), 113(103), 307(101), 331(109), 601(199), 1303(433), 19(5), 457(151), 421(139), 547(181), 2221(739), 691(229), 757(251), 3967(1321), 727(241), 163(53), 3517(1171), 1063(353), 1783(593), 1987(661), 1861(619), 271(89), 6271(2089), 997(331), 1123(373), 6163(2053).

Semiprimes obtained and the corresponding p in the brackets: 5^2(7), 5*41(67), 5*23(37), 11*17(61), 5*11(17), 5*347(577), 17*29(163), 19*43(271), 11*233(853), 13*229(991), 5*239(397), 53*101(1783), 37*241(2971), 11*53(193), 13*151(653), 23*41(313), 5*563(937), 31*43(443).

Numbers divisible by 5 (not semiprimes) obtained and the corresponding p in the brackets: 5^2*37(307), 5^2*109(907), 5^2*487(4057), 5^2*19(157), 5*13*179(3877).

This formula produces 35 primes for the first 55 values of p!

The formula can be extrapolated for all Carmichael numbers and all their prime factors: primes of type 3*p + d - 3, where p is a prime factor of a Carmichael number divisible by d; for instance, were obtained the following primes of type 3*p + 10, where p is a prime factor of a Carmichael number divisible by 13: 61, 31, 67, 103, 193, 43, 229, 337, 1201, 79, 211, 823, 607, 463, 1741, 499, 643, 733, 97, 2029, 139, 349, 4129, 6421, 1381, 2731, 1069, 853, 1021, 9421, 5413, 10831, 223, 1933, 8269 (which means 35 primes) for the first 55 values of p!

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Carmichael Number

Crossrefs

Cf. A002997.