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0, 1, 2, 4, 5, 6, 7, 11, 13, 17, 19, 21, 26, 27, 31, 35, 37, 40, 43, 47, 49, 51, 57, 66, 73, 79, 81, 93, 95, 109, 111, 113, 119, 120, 127, 129, 133, 153, 155, 163, 172, 173, 177, 185, 189, 211, 213, 223, 245, 247, 253, 271, 277, 279, 283, 289, 301, 303, 309, 336
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OFFSET
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1,3
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COMMENTS
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Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence lists the records in A300858, which is a function that represents the difference between the latter and the former species of nondivisors in the cototient of n.
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LINKS
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EXAMPLE
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0 is the first term since A300858(1) = 0. A300858 is 0 or negative for n < 8.
A300858(8) = A243823(8) - A243822(8) = 1 - 0 = 1. Within the cototient of 8 there is one nondivisor (6) and it does not divide 8^e for integer e. (All prime powers m have A243822(m) = 0 and for m > 4, A243823(m) is positive.) Therefore 1 is the next term. Between 8 and 15, -1 <= A300858(n) <= 1.
A300858(15) = 2. Within the cototient of 15 there are 4 nondivisors; of these 3 (i.e., {6, 10, 12}) do not divide 15^e for integer e, but 9 | 15^2. Therefore 3 - 1 = 2 and 2 exceeds all values A300858(n) for n < 15, and appears after 1.
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MATHEMATICA
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f[n_] := Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]; Union@ FoldList[Max, Array[#1 - #3 + 1 - 2 #2 + #4 & @@ {#, f@ #, EulerPhi@ #, DivisorSigma[0, #]} &, 600]]
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PROG
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(PARI) a300858(n) = 1 + n + numdiv(n) - eulerphi(n) - 2*sum(k=1, n, if(gcd(n, k)-1, 0, moebius(k)*(n\k))) \\ after Michel Marcus in A300858
r=-1; for(x=1, oo, if(a300858(x) > r, r=a300858(x); print1(r, ", "))) \\ Felix Fröhlich, Mar 30 2018
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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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