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A300859
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Where records occur in A045763.
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3
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1, 6, 10, 14, 18, 22, 26, 30, 36, 38, 42, 50, 54, 60, 66, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 168, 174, 180, 186, 198, 204, 210, 234, 240, 246, 252, 258, 264, 270, 294, 300, 318, 330, 360, 378, 390, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600
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history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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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.
A045763(p) = 0 for p prime, therefore there are no primes in a(n).
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).
Conjectures based on 1255 terms of a(n) < 36,000,000:
1. There are no prime powers p^e > 1 in a(n), i.e., the intersection of a(n) and A000961 is {1}.
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.
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.
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.
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, p. 352 (sixth edition), see Theorem 327.
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LINKS
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FORMULA
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EXAMPLE
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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.
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.
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MATHEMATICA
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With[{s = Array[1 + # - EulerPhi@ # - DivisorSigma[0, #] &, 10^3]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]
DeleteDuplicates[Table[{n, n+1-DivisorSigma[0, n]-EulerPhi[n]}, {n, 600}], GreaterEqual[#1 [[2]], #2 [[2]]]&][[;; , 1]] (* Harvey P. Dale, Mar 29 2023 *)
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CROSSREFS
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Cf. A000005, A000010, A002110, A002182, A045763, A133995, A243822, A243823, A272618, A272619, A293555, A292867, A300156, A300858, A300861.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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