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A294492
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Numbers m that set records for the ratio A045763(n)/n.
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2
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1, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 90, 102, 114, 126, 138, 150, 210, 330, 390, 420, 510, 570, 630, 1050, 1470, 2310, 4620, 6930, 11550, 16170, 25410, 30030, 60060, 90090, 150150, 210210, 330330, 390390, 510510, 1021020, 1531530, 2552550, 3573570
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OFFSET
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1,2
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COMMENTS
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These numbers have an increasing proportion of nondivisors in the cototient (A051953(n)) with respect to n.
In other words, these numbers have an increasing proportion of smaller numbers that are counted neither by tau or phi.
Conjectures:
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).
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).
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LINKS
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EXAMPLE
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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.
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).
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.
Table of terms less than A002110(6):
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.
n a(n) b(n) c(n)
1 1 0 0
2 6 1 1 1
3 10 3 1 0 1
4 14 5 1 0 0 1
5 18 7 1 2
6 22 9 1 0 0 0 1
7 26 11 1 0 0 0 0 1
8 30 15 1 1 1
9 42 23 1 1 0 1
10 60 33 2 1 1
11 66 39 1 1 0 0 1
12 78 47 1 1 0 0 0 1
13 90 55 1 2 1
14 102 63 1 1 0 0 0 0 1
15 114 71 1 1 0 0 0 0 0 1
16 126 79 1 2 0 1
17 138 87 1 1 0 0 0 0 0 0 1
18 150 99 1 1 2
19 210 147 1 1 1 1
20 330 235 1 1 1 0 1
21 390 279 1 1 1 0 0 1
22 420 301 2 1 1 1
23 510 367 1 1 1 0 0 0 1
24 570 411 1 1 1 0 0 0 0 1
25 630 463 1 2 1 1
26 1050 787 1 1 2 1
27 1470 1111 1 1 1 2
28 2310 1799 1 1 1 1 1
29 4620 3613 2 1 1 1 1
30 6930 5443 1 2 1 1 1
31 11550 9103 1 1 2 1 1
32 16170 12763 1 1 1 2 1
33 25410 20083 1 1 1 1 2
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MAPLE
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with(numtheory): P:=proc(q) local a, b, n; a:=-1; for n from 1 to q do
b:=n+1-tau(n)-phi(n); if b>a then a:=b; print(n); fi; od; end: P(10^2);
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MATHEMATICA
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With[{s = Array[(# - (DivisorSigma[0, #] + EulerPhi@ # - 1))/# &, 10^6]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]]
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CROSSREFS
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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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