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A048868
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Numbers for which reduced residue system contains more primes than nonprimes.
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5
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8, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170
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OFFSET
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1,1
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COMMENTS
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Conjecture: all terms are even.
Let b(n) = differences of a(n). Many of the terms in b(n) are in A060735(n), i.e., b(n) >= A002110(n) and divisible by A002110(n). The smallest b(n) that is not in A060735 is b(450) = 66; b(450) > A002110(n) but not divisible by it; instead it is divisible by A002110(n-1). b(n) seems to "prefer" values of A002110(n) as n increases. There is a run of 22 values of 2 starting at b(2), of 22 values of 6 starting at b(178), of 7 values of 210 starting at b(543), and 3 values of 2310 starting at b(577). In the case of A002110(n) with n < 3, the length of runs of A002110(n) is greater than that of runs of any other number of comparable magnitude. The last observed position of A002110(n) in b(n) is {201, 442, (557), ...}.
Questions: are there increasingly more terms such that b(n) is not in A060735 as n increases? Are terms such that b(n) is in A060735 prevalent? Why does b(n) seem to "prefer" values in A002110 at certain magnitudes of n as n increases? Are there differences of a(n) = A002110(k) at positions greater than those observed, and if not, why? (End)
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LINKS
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EXAMPLE
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n=30 is the largest extremal example whose reduced residue system consists only of primes and 1 (see A048597); n=8 Phi(8)=4, reduced residue system (8)={1,3,5,7} n=32 Phi(32)=16, reduced residue system (32)={1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31} of which only {1,9,15,21,25,27} are not primes,10 are primes: 10>6 thus 32 belongs here.
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MATHEMATICA
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Select[Range@ 170, Function[n, Count[Range@ n, _?(PrimeQ@ # && CoprimeQ[n, #] &)] > EulerPhi[n]/2]] (* Michael De Vlieger, May 21 2017 *)
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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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