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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

Sequence in context: A299929 A287217 A101324 * A167462 A088513 A004611

Adjacent sequences:  A216827 A216828 A216829 * A216831 A216832 A216833

KEYWORD

nonn

AUTHOR

Marius Coman, Sep 19 2012

STATUS

approved

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Last modified August 4 13:49 EDT 2020. Contains 336201 sequences. (Running on oeis4.)