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A300860
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Indices of records in A300858.
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2
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1, 8, 15, 16, 26, 27, 28, 32, 44, 52, 56, 62, 64, 76, 80, 88, 96, 100, 104, 112, 122, 124, 128, 144, 160, 176, 184, 192, 200, 216, 246, 248, 250, 256, 272, 276, 282, 288, 318, 320, 324, 348, 354, 366, 372, 384, 414, 426, 432, 468, 474, 486, 516, 522, 528, 534
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OFFSET
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1,2
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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 record setters in the sequence A300858(n), which is a function that represents the difference between the latter and the former species of nondivisors in the cototient of n.
Odd terms m < 36,000,000: {1, 15, 27}.
Smallest term m with A001221(m) = {0, 1, 2, ..., 8} = {1, 8, 15, 246, 2010, 9870, 30030, 510510, 9699690} (the last 3 terms are in A002110).
Smallest term m with A001222(m) = {0, 2, 3, ..., 12} = {1, 15, 8, 16, 32, 64, 128, 256, 768, 1536, 7680, 53760, 3843840} (includes 2^e with 3 <= e <= 8). Note, A300858(p) for p prime = 0.
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LINKS
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EXAMPLE
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8 is in the sequence because A300858(n) for n < 8 is negative or 0 after A300858(1) = 0. 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.)
15 is in the sequence because -1 <= A300858(n) <= 1 for n < 15. 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.
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MATHEMATICA
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f[n_] := Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]; With[{s = Array[#1 - #3 + 1 - 2 #2 + #4 & @@ {#, f@ #, EulerPhi@ #, DivisorSigma[0, #]} &, 550]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] ]
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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
r=-1; for(i=1, oo, if(a300858(i) > r, print1(i, ", "); r=a300858(i))) \\ 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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