A299992 -id:A299992 - cp's OEIS frontend

A299990 a(n) = A243822(n) - A000005(n).

-1, -2, -2, -3, -2, -3, -2, -4, -3, -2, -2, -4, -2, -2, -3, -5, -2, -2, -2, -4, -3, -1, -2, -5, -3, -1, -4, -4, -2, 2, -2, -6, -2, 0, -3, -4, -2, 0, -2, -5, -2, 3, -2, -3, -4, 0, -2, -5, -3, 0, -2, -3, -2, 0, -3, -5, -2, 0, -2, 2, -2, 0, -4, -7, -3, 6, -2, -2

Offset: 1

Michael De Vlieger, Feb 25 2018

sign

Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among "regular" numbers.
Value of a(n) is generally less frequently negative as n increases.
a(1) = -1.
For primes p, a(p) = -2 since 1 | p and the cototient is restricted to the divisor p.
For perfect prime powers p^e, a(p^e) = -(e + 1), since all m < p^e in the cototient of p^e that do not have a prime factor q coprime to p^e are powers p^k with 1 < p^k <= p^e; all such p^k divide p^e.
Generally for n with A001221(n) = 1, a(n) = -1 * A000005(n), since the cototient is restricted to divisors, and in the case of p^e > 4, divisors and numbers in A272619(p^e) not counted by A010846(p^e).
For m >= 3, a(A002110(m)) is positive.
For m >= 9, a(A244052(m)) is positive.

			a(6) = -3 since 6 has 4 divisors, and 4 | 6^2; A243822(6) = 1 and A000005(6) = 4; 1 - 4 = -3. Alternatively, A010846(6) = 5; 5 - 2*4 = -3.
a(30) = 2 since 30 has 8 divisors and the numbers {4, 8, 9, 12, 16, 18, 20, 24, 25, 27} divide 30^e with e > 1; A243822(30) = 10 and A000005(30) = 8; 10 - 8 = 2. Alternatively, A010846(30) = 18; 18 - 2*8 = 2.
Some values of a(n) and related sequences:
   n  a(n) A010846(n) A243822(n) A000005(n) A272618(n)
  ----------------------------------------------------
   1   -1          1          0          1  0
   2   -2          2          0          2  0
   3   -2          2          0          2  0
   4   -3          3          0          3  0
   5   -2          2          0          2  0
   6   -3          5          1          4  {4}
   7   -2          2          0          2  0
   8   -4          4          0          4  0
   9   -3          3          0          3  0
  10   -2          6          2          4  {4,8}
  11   -2          2          0          2  0
  12   -4          8          2          6  {8,9}
  ...
  30    2         18         10          8  {4,8,9,12,16,18,20,24,25,27}
  ...
  34    0          8          4          4  {4,8,16,32}
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Examination of the relationships of the species of numbers enumerated in A010846.

Cf. A000005, A002110, A010846, A243822, A272618, A272619, A299991, A299992, A300155, A300156, A300157.

Table[Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] - 2 DivisorSigma[0, n], {n, 68}]

a(n) = A010846(n) - 2*A000005(n).

A300155 Numbers n for which A243822(n) = A000005(n).

Original entry on oeis.org

34, 38, 46, 50, 54, 58, 62, 105, 249, 267, 268, 284, 291, 292, 303, 309, 316, 321, 324, 327, 332, 339, 356, 363, 381, 385, 388, 393, 404, 411, 412, 417, 428, 436, 447, 452, 453, 455, 471, 484, 489, 500, 501, 507, 508, 519, 537, 543, 573, 579, 591, 595, 597

Offset: 1

Michael De Vlieger, Feb 26 2018

nonn

Indices of zeros in A299990, i.e., A010846(n) - 2*A000005(n) = 0.
Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A045763(n). Divisors of n are listed in row n of A027750.
This sequence lists numbers that have an equal number of nondivisors k in the cototient of n as divisors d.
The smallest odd term is 105.

			34 is the first term since it is the smallest number for which A243822(34) = A000005(34). For n = 34, there are 4 divisors {1, 2, 17, 34} and 4 nondivisors 1 <= m <= n such that m | n^e with e > 1: {4, 8, 16, 32}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A010846, A027750, A045763, A162306, A243822, A272618, A299990, A299991, A299992.

Select[Range@ 600, Function[n, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] == 2 DivisorSigma[0, n]]]

A300156 Indices of records in A299990.

Original entry on oeis.org

1, 30, 42, 66, 78, 90, 102, 114, 138, 150, 210, 330, 390, 510, 570, 630, 870, 990, 1050, 1470, 1890, 2100, 2310, 2730, 3570, 3990, 4620, 5460, 6510, 6930, 8190, 9240, 10710, 11550, 13650, 16170, 19110, 20790, 23100, 24570, 25410, 30030, 39270, 43890, 46410

Offset: 1

Michael De Vlieger, Feb 26 2018

nonn

A010846(n) = A000005(n) + A243822(n).
Successive terms in this sequence represent increasing differences A243822(n) - A000005(n).
A000079 = records in -1 * A299990, since A243822(p^e)=0 for e>=0, n = 2^k sets records in A000005(n). The corresponding records are in A000027.

			The first term is 1: A299990(1) = -1. The first term that exceeds -1 in A299990 corresponds to n = 30; A299990(30) = 2, thus 30 is the second term.

Michael De Vlieger, Table of n, a(n) for n = 1..108
Michael De Vlieger, Relationships between A244052, A294492, and A300156.

Cf. A000005, A000027, A000079, A010846, A243822, A299990, A299991, A299992, A300155, A300157.

With[{s = Table[Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] - 2 DivisorSigma[0, n], {n, 10^3}]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]

A300157 Records in A299990.

Original entry on oeis.org

-1, 2, 3, 6, 7, 8, 9, 10, 11, 17, 36, 45, 48, 54, 56, 67, 69, 76, 97, 118, 119, 120, 219, 231, 249, 258, 286, 299, 302, 356, 367, 377, 392, 455, 471, 533, 547, 563, 573, 576, 647, 1033, 1096, 1125, 1129, 1157, 1178, 1334, 1405, 1439, 1587, 1664, 1721, 1952

Offset: 1

Michael De Vlieger, Feb 26 2018

sign

A010846(n) = A000005(n) + A243822(n).
A000079 = records in -1 * A299990, since A243822(p^e)=0 for e>=0, n = 2^k sets records in A000005(n). The corresponding records are in A000027.
Number of terms less than 10^k with 0 <= k <= 7: {0, 1, 6, 18, 32, 51, 68, 96}.

			A299990(1) = -1, thus this is the first record. A299990(30) = 2 and is the first term greater than -1, thus 30 is the second term of this sequence. Observe that A243822(1) is 0 and A000005(1) is 1; 0 - 1 = -1. A243822(30) is 10 and A000005(30) = 8; 10 - 8 = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..108

Cf. A000005, A000027, A000079, A010846, A243822, A299990, A299991, A299992, A300155, A300156.

Union@ FoldList[Max, Table[Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] - 2 DivisorSigma[0, n], {n, 10^3}]]

cp's OEIS Frontend

A299990 a(n) = A243822(n) - A000005(n).

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A300155 Numbers n for which A243822(n) = A000005(n).

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A300156 Indices of records in A299990.

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A300157 Records in A299990.

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