%I #11 Apr 03 2018 16:07:19
%S 1,8,15,16,26,27,28,32,44,52,56,62,64,76,80,88,96,100,104,112,122,124,
%T 128,144,160,176,184,192,200,216,246,248,250,256,272,276,282,288,318,
%U 320,324,348,354,366,372,384,414,426,432,468,474,486,516,522,528,534
%N Indices of records in A300858.
%C 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.
%C Odd terms m < 36,000,000: {1, 15, 27}.
%C 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).
%C 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.
%H Michael De Vlieger, <a href="/A300860/b300860.txt">Table of n, a(n) for n = 1..1710</a>
%H Michael De Vlieger, <a href="/A300860/a300860.txt">Decomposition of terms in A300860 and Related Sequences.</a>
%e 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.)
%e 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.
%t 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]] ]
%o (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_
%o r=-1; for(i=1, oo, if(a300858(i) > r, print1(i, ", "); r=a300858(i))) \\ _Felix Fröhlich_, Mar 30 2018
%Y Cf. A001221, A001222, A243822, A243823, A300858.
%K nonn
%O 1,2
%A _Michael De Vlieger_, Mar 14 2018