A300860 Indices of records in A300858.

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

Offset: 1

Author

Michael De Vlieger, Mar 14 2018

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1710
• Michael De Vlieger, Decomposition of terms in A300860 and Related Sequences.

Crossrefs

Cf. A001221, A001222, A243822, A243823, A300858.

Programs

Mathematica

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]] ]

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