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 A300859 #14 Mar 29 2023 14:47:47
%S A300859 1,6,10,14,18,22,26,30,36,38,42,50,54,60,66,78,84,90,102,114,120,126,
%T A300859 132,138,150,168,174,180,186,198,204,210,234,240,246,252,258,264,270,
%U A300859 294,300,318,330,360,378,390,420,450,462,480,504,510,540,546,570,600
%N A300859 Where records occur in A045763.
%C A300859 The cototient of n consists of numbers 1 < m <= n that are not coprime to n, i.e., gcd(m,n) > 1. These numbers have at least one prime divisor p that also divides n. The cototient of n contains the divisors d of n; the remaining nondivisors in the cototient of n are listed in A133995. The counting function of A133995 is A045763(n). There are two species of numbers in the nondivisor-cototient of n: those in row n of A272618, of which A243822(n) is counting function, and those in row n of A272619, of which A243823(n) is the counting function. The former species divides n^e for integer e > 1, while the latter does not divide any integer power of n.
%C A300859 A045763(p) = 0 for p prime, therefore there are no primes in a(n).
%C A300859 Except for prime terms (i.e., 2), A002110 is a subset as primorials minimize the totient function. The divisor counting function is increasingly vanishingly small compared to the totient function for A002110(i) as i increases, and A002110(i) for 1 < i <= 9 is observed in a(n).
%C A300859 Conjectures based on 1255 terms of a(n) < 36,000,000:
%C A300859 1. There are no prime powers p^e > 1 in a(n), i.e., the intersection of a(n) and A000961 is {1}.
%C A300859 2. A293555 is a subset of A300859. Numbers that have a lot of nondivisors m | n^e with e > 1 (i.e., in row n of A272618 and counted by A243822(n)) tend to reduce the totient and increasingly have fewer divisors than highly composite numbers, widening the nondivisor-cototient.
%C A300859 3. A300156 is a subset of A300859. Numbers that have more nondivisors m | n^e with e > 1 (i.e., in row n of A272618 and counted by A243822(n)) than divisors tend to reduce the totient and have fewer divisors than highly composite numbers (i.e., those n in A002182), widening the nondivisor-cototient.
%C A300859 Increasingly many terms k in A262867 also appear in a(n) as k increases. A292867 lists record-setters in A243823, which is the counting function of one of the two species of nondivisors in the cototient of n.
%D A300859 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, p. 352 (sixth edition), see Theorem 327.
%H A300859 Michael De Vlieger, <a href="/A300859/b300859.txt">Table of n, a(n) for n = 1..1735</a>
%H A300859 Michael De Vlieger, <a href="/A300859/a300859.txt">Decomposition of terms in A300859 and Related Sequences.</a>
%F A300859 a(n) = 1 + n - A000010(n) - A000005(n).
%e A300859 6 is in the sequence because there is 1 nondivisor in the cototient of 6 (i.e., 4), and that total exceeds 0 for all smaller positive numbers.
%e A300859 10 follows 6 because there are 3 nondivisors in the cototient (4, 6, 8), and this exceeds the total of 1 for n = 6, 8, and 9.
%t A300859 With[{s = Array[1 + # - EulerPhi@ # - DivisorSigma[0, #] &, 10^3]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]
%t A300859 DeleteDuplicates[Table[{n,n+1-DivisorSigma[0,n]-EulerPhi[n]},{n,600}],GreaterEqual[#1 [[2]],#2 [[2]]]&][[;;,1]] (* _Harvey P. Dale_, Mar 29 2023 *)
%Y A300859 Cf. A000005, A000010, A002110, A002182, A045763, A133995, A243822, A243823, A272618, A272619, A293555, A292867, A300156, A300858, A300861.
%K A300859 nonn
%O A300859 1,2
%A A300859 _Michael De Vlieger_, Mar 15 2018