cp's OEIS Frontend

A347406 Earliest sequence of distinct positive integers such that both gcd(a(n),a(n-k)) = 1 and gcd(a(n),a(n+k)) = 1, where k is each divisor of a(n) and n - k >= 1.

%I A347406 #11 Sep 07 2021 13:50:41
%S A347406 1,2,3,5,4,7,9,8,11,13,10,17,19,14,15,23,16,29,21,26,27,25,22,31,35,
%T A347406 32,33,37,38,41,39,34,43,47,28,53,51,20,57,59,40,61,49,44,63,67,46,71,
%U A347406 73,52,69,79,50,83,81,55,58,77,65,82,87,85,89,74,93,95,91,86,97,101,62,103,45,64,75
%N A347406 Earliest sequence of distinct positive integers such that both gcd(a(n),a(n-k)) = 1 and gcd(a(n),a(n+k)) = 1, where k is each divisor of a(n) and n - k >= 1.
%C A347406 The majority of terms are concentrated along two lines, the upper line has gradient of approximately 1.37 while the lower line has a gradient of approximately 1.02. Between these a third more random line also appear. See the linked image.
%C A347406 Small numbers with only 2 and 3 as prime divisors apparently take many terms to appear. For example a(210613) = 6, a(224221) = 18, while 12 and 24 have not appeared after 250000 terms.
%H A347406 Scott R. Shannon, <a href="/A347406/a347406.png">Image of the first 250000 terms</a>.
%e A347406 a(3) = 3 as the divisors of 3 are 1 and 3, and a(3-1) = a(2) = 2, a(3+1) = a(4) = 5, and a(3+3) = a(6) = 7, and the gcd of 3 and each of these three numbers is 1. As a(3-3) = a(0) is not defined this term is ignored.
%e A347406 a(11) = 10 as the divisors of 10 are 1, 2, 5 and 10, and a(11-1) = a(10) = 13, a(11-2) = a(9) = 11, a(11-5) = a(6) = 7, a(11-10) = a(1) = 1, a(11+1) = a(12) = 17, a(11+2) = a(13) = 19, a(11+5) = a(16) = 23, and a(11+10) = a(21) = 27, and the gcd of 10 and each of these eight numbers is 1.
%e A347406 a(13) = 19 as the divisors of 19 are 1 and 19, and a(13-1) = a(12) = 17, a(13+1) = a(14) = 14, and a(13+19) = a(32) = 34, and the gcd of 19 and each of these three numbers is 1. Note that as a(11) = 10, and a(11+2) = a(13), where 2 is a divisor of 10, a(13) cannot equal 15 as gcd(10,15) > 1. This is the first term that differs from A347179.
%Y A347406 Cf. A347179, A027750, A000005, A084937, A098550, A336957.
%K A347406 nonn,look
%O A347406 1,2
%A A347406 _Scott R. Shannon_, Aug 30 2021