cp's OEIS Frontend

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.

%I A216830 #21 Feb 16 2025 08:33:18
%S A216830 7,13,19,31,23,41,67,73,11,103,37,101,61,109,199,433,5,17,151,577,307,
%T A216830 163,139,181,271,739,229,251,853,1321,991,241,53,397,1783,1171,907,
%U A216830 2971,353,593,4057,661,193,619,89,653,157,2089,313,331,373,937,2053,443,3877
%N 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.
%C A216830 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.
%C A216830 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).
%C A216830 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).
%C A216830 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).
%C A216830 This formula produces 35 primes for the first 55 values of p!
%C A216830 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!
%H A216830 Charles R Greathouse IV, <a href="/A216830/b216830.txt">Table of n, a(n) for n = 1..10000</a>
%H A216830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%Y A216830 Cf. A002997.
%K A216830 nonn
%O A216830 1,1
%A A216830 _Marius Coman_, Sep 19 2012