A294492 Numbers m that set records for the ratio A045763(n)/n.

1, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 90, 102, 114, 126, 138, 150, 210, 330, 390, 420, 510, 570, 630, 1050, 1470, 2310, 4620, 6930, 11550, 16170, 25410, 30030, 60060, 90090, 150150, 210210, 330330, 390390, 510510, 1021020, 1531530, 2552550, 3573570

Offset: 1

Author

Michael De Vlieger, Nov 01 2017

Keywords

• nonn

Comments

These numbers have an increasing proportion of nondivisors in the cototient (A051953(n)) with respect to n.

In other words, these numbers have an increasing proportion of smaller numbers that are counted neither by tau or phi.

Conjectures:

1. Let k = any product of primorial A002110(i - 1) and the smallest i primes. All terms m are in A002110 or of the form k*p, with prime p >= prime(i) such that k < A002110(i + 1).

2. For m >= A002110(5) = 2310, all terms m are in A002110 or of the form prime p * A002110(i), with prime(1) <= p <= prime(i).

Examples

			1 is in the sequence since 1 is coprime to and a divisor of all numbers, therefore it has no nondivisors in the cototient, i.e., A045763(1)/1 = 0. The primes have no nondivisors in the cototient, 4 only has divisors in the cototient.
6 has the nondivisor 4 in the cototient, thus 1/6, thus it appears after 1 in the sequence. The following numbers do not appear, as 7 has none, 8 has one (6), 9 has one (6).
10 has the nondivisors (4,6,8) in the cototient, thus 3/10. Since 3/10 > 1/6, 10 is the next number in the sequence.
Table of terms less than A002110(6):
b(n) = A045763(n), c(n) = exponents of the smallest primes such that the product = n, e.g., "2 1 0 1" = 2^2 * 3^1 * 5^0 * 7^1 = 126.
   n    a(n)   b(n) c(n)
   1      1      0  0
   2      6      1  1 1
   3     10      3  1 0 1
   4     14      5  1 0 0 1
   5     18      7  1 2
   6     22      9  1 0 0 0 1
   7     26     11  1 0 0 0 0 1
   8     30     15  1 1 1
   9     42     23  1 1 0 1
  10     60     33  2 1 1
  11     66     39  1 1 0 0 1
  12     78     47  1 1 0 0 0 1
  13     90     55  1 2 1
  14    102     63  1 1 0 0 0 0 1
  15    114     71  1 1 0 0 0 0 0 1
  16    126     79  1 2 0 1
  17    138     87  1 1 0 0 0 0 0 0 1
  18    150     99  1 1 2
  19    210    147  1 1 1 1
  20    330    235  1 1 1 0 1
  21    390    279  1 1 1 0 0 1
  22    420    301  2 1 1 1
  23    510    367  1 1 1 0 0 0 1
  24    570    411  1 1 1 0 0 0 0 1
  25    630    463  1 2 1 1
  26   1050    787  1 1 2 1
  27   1470   1111  1 1 1 2
  28   2310   1799  1 1 1 1 1
  29   4620   3613  2 1 1 1 1
  30   6930   5443  1 2 1 1 1
  31  11550   9103  1 1 2 1 1
  32  16170  12763  1 1 1 2 1
  33  25410  20083  1 1 1 1 2
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..53
• Michael De Vlieger, Numbers m that set records for the ratio A045763(n)/n.

Crossrefs

Cf. A002110, A045763, A051953.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,n; a:=-1; for n from 1 to q do
b:=n+1-tau(n)-phi(n); if b>a then a:=b; print(n); fi; od; end: P(10^2);
# Paolo P. Lava, Nov 17 2017

Mathematica

With[{s = Array[(# - (DivisorSigma[0, #] + EulerPhi@ # - 1))/# &, 10^6]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]]