|
%I #8 Jun 26 2018 06:50:18
%S 277,1051,1381,1657,1867,3001,3373,3499,4637,4877,5147,6673,7547,
%T 10859,10987,14797,17291,18749,19531,25939,27337,27953,31013,32203,
%U 32983,33547,34123,34591,35747,38047,38197,38711,44293,44357,47059,47569,48809,51151,51437
%N Numbers that divide exactly two Euclid numbers.
%C The k-th Euclid number, A006862(k), is 1 plus the product of the first k primes, i.e., 1 + A002110(k). A113165 lists the numbers (> 1) that divide at least one Euclid number; a(1) = 277 = A113165(19); a(2) = 1051 = A113165(41); a(53) = 92143 = A113165(995).
%C Up to N = 10^5, roughly 5% of the terms in A113165 are also in this sequence. Does that ratio continue to hold as N increases?
%C It appears that the vast majority of terms in A113165 are prime, but that sequence contains a number of composite numbers as well, beginning with A113165(59) = 1843 = 19*97, A113165(125) = 5263 = 19*277, A113165(195) = 10147 = 73*139, and A113165(231) = 12629 = 73*173. But do any composites divide more than one Euclid number?
%H Giovanni Resta, <a href="/A297891/b297891.txt">Table of n, a(n) for n = 1..1000</a>
%e a(1) = 277 because 277 is the smallest number that divides exactly two Euclid numbers: 1 + 2*3*5*7*11*13*17 = 510511 and 1 + 2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59 = 1922760350154212639071.
%Y Cf. A002110 (primorials), A006862 (Euclid numbers), A113165 (numbers > 1 that divide Euclid numbers).
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jan 07 2018
|
