This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A294492 #19 Nov 22 2017 03:18:51
%S A294492 1,6,10,14,18,22,26,30,42,60,66,78,90,102,114,126,138,150,210,330,390,
%T A294492 420,510,570,630,1050,1470,2310,4620,6930,11550,16170,25410,30030,
%U A294492 60060,90090,150150,210210,330330,390390,510510,1021020,1531530,2552550,3573570
%N A294492 Numbers m that set records for the ratio A045763(n)/n.
%C A294492 These numbers have an increasing proportion of nondivisors in the cototient (A051953(n)) with respect to n.
%C A294492 In other words, these numbers have an increasing proportion of smaller numbers that are counted neither by tau or phi.
%C A294492 Conjectures:
%C A294492 1. Let k = any product of primorial A002110(i - 1) and the smallest i primes. All terms m are in A002110 or of the form k*p, with prime p >= prime(i) such that k < A002110(i + 1).
%C A294492 2. For m >= A002110(5) = 2310, all terms m are in A002110 or of the form prime p * A002110(i), with prime(1) <= p <= prime(i).
%H A294492 Michael De Vlieger, <a href="/A294492/b294492.txt">Table of n, a(n) for n = 1..53</a>
%H A294492 Michael De Vlieger, <a href="/A294492/a294492.txt">Numbers m that set records for the ratio A045763(n)/n.</a>
%e A294492 1 is in the sequence since 1 is coprime to and a divisor of all numbers, therefore it has no nondivisors in the cototient, i.e., A045763(1)/1 = 0. The primes have no nondivisors in the cototient, 4 only has divisors in the cototient.
%e A294492 6 has the nondivisor 4 in the cototient, thus 1/6, thus it appears after 1 in the sequence. The following numbers do not appear, as 7 has none, 8 has one (6), 9 has one (6).
%e A294492 10 has the nondivisors (4,6,8) in the cototient, thus 3/10. Since 3/10 > 1/6, 10 is the next number in the sequence.
%e A294492 Table of terms less than A002110(6):
%e A294492 b(n) = A045763(n), c(n) = exponents of the smallest primes such that the product = n, e.g., "2 1 0 1" = 2^2 * 3^1 * 5^0 * 7^1 = 126.
%e A294492 n a(n) b(n) c(n)
%e A294492 1 1 0 0
%e A294492 2 6 1 1 1
%e A294492 3 10 3 1 0 1
%e A294492 4 14 5 1 0 0 1
%e A294492 5 18 7 1 2
%e A294492 6 22 9 1 0 0 0 1
%e A294492 7 26 11 1 0 0 0 0 1
%e A294492 8 30 15 1 1 1
%e A294492 9 42 23 1 1 0 1
%e A294492 10 60 33 2 1 1
%e A294492 11 66 39 1 1 0 0 1
%e A294492 12 78 47 1 1 0 0 0 1
%e A294492 13 90 55 1 2 1
%e A294492 14 102 63 1 1 0 0 0 0 1
%e A294492 15 114 71 1 1 0 0 0 0 0 1
%e A294492 16 126 79 1 2 0 1
%e A294492 17 138 87 1 1 0 0 0 0 0 0 1
%e A294492 18 150 99 1 1 2
%e A294492 19 210 147 1 1 1 1
%e A294492 20 330 235 1 1 1 0 1
%e A294492 21 390 279 1 1 1 0 0 1
%e A294492 22 420 301 2 1 1 1
%e A294492 23 510 367 1 1 1 0 0 0 1
%e A294492 24 570 411 1 1 1 0 0 0 0 1
%e A294492 25 630 463 1 2 1 1
%e A294492 26 1050 787 1 1 2 1
%e A294492 27 1470 1111 1 1 1 2
%e A294492 28 2310 1799 1 1 1 1 1
%e A294492 29 4620 3613 2 1 1 1 1
%e A294492 30 6930 5443 1 2 1 1 1
%e A294492 31 11550 9103 1 1 2 1 1
%e A294492 32 16170 12763 1 1 1 2 1
%e A294492 33 25410 20083 1 1 1 1 2
%p A294492 with(numtheory): P:=proc(q) local a,b,n; a:=-1; for n from 1 to q do
%p A294492 b:=n+1-tau(n)-phi(n); if b>a then a:=b; print(n); fi; od; end: P(10^2);
%p A294492 # _Paolo P. Lava_, Nov 17 2017
%t A294492 With[{s = Array[(# - (DivisorSigma[0, #] + EulerPhi@ # - 1))/# &, 10^6]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]]
%Y A294492 Cf. A002110, A045763, A051953.
%K A294492 nonn
%O A294492 1,2
%A A294492 _Michael De Vlieger_, Nov 01 2017