A287917 Number of distinct primorials A002110(k) > A285784(n) such that the primorials are coprime to A285784(n).

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

Offset: 2

Author

Jamie Morken and Michael De Vlieger, Jun 08 2017

Keywords

• nonn

Comments

Terms of A285784 that have a(n) = 1 appear in A287390.

Terms of A285784 that have a(n) > 1 appear in A287391.

From Michael De Vlieger, Jun 09 2017: (Start)

Let primorial p_n# = A002110(n) and let m be a nonzero positive number called a totative such that gcd(t, p_n#) = 1. This sequence concerns nonprime m. A285784 is the sequence that lists unique nonprime totatives m of primorials p_n#.

For A285784(1), a(n) = infinity, since 1 is the empty product and a totative of (i.e., coprime to) all numbers. Hence the offset of a(n) is 2 and for this reason hereinafter we only consider composite totatives m.

Consider the composite totative m in A285784. For a given composite term in A285784, there is a least primorial p_a# to which m is coprime. Such m < p_a# are products of prime totatives q > p_a, the gpf of p_a#. Therefore m "appears" when there are prime totatives q < sqrt(p_a#). The smallest a for which we have this condition is a = 4, as q = 11 is less than sqrt(210). For the same reason the first composite term of A285784 is 11^2 = 121.

For n >= 2, m is coprime to a finite range of primorials p_a# .. p_b#. If m is coprime to p_b#, then it must be coprime to all primorials p_a# .. p_b# by the definition of primorial. m is no longer coprime to p_(b+1)# since at least one of its prime divisors p_(b+1) also divides p_(b+1)#. This sequence gives the range b - a + 1.

To generate data that includes all the terms of A285784 less than a limit x, we can write a while statement that operates so long as there is at least 1 totative m < x of p_n#. Since primorial p_n# is the product of the smallest n primes, fewer numbers less than x are coprime to p_n# as n increases, until exhaustion. Thus we can produce a list of unique m < x (i.e., terms of A285784 less than x) for relatively large primorials p_n#. Then we can count the instances of terms of A285784 for a list of lists of totatives m < x for primorials p_1# .. p_n# and obtain certainty about the number of instances of terms of A285784.

First position of values of a(n): {2, 4, 12, 20, 38, 47, 76, 96, 111, 139, 228, 241, 300, 339, 363, 434, 482, 566, 689, 752, 790, 862, 902, 973, 1264, 1361, 1506, 1562, 1816, ...}

Terms of A285784 that set records in a(n): {121, 169, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, ...}

(End)

Examples

			The sequence starts:
   n  A285784(n)  a(n)
   2      121     1
   3      143     1
   4      169     2
   5      187     1
   6      209     1
   7      221     1
   8      247     1
   9      289     2
  10      299     1
  11      323     2
  12      361     3
  13      377     1
  14      391     2
  15      403     1
  16      437     3
  17      481     1
  18      493     2   ...
		

Links

• Jamie Morken, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A002110, A285784, A287390, A287391.

Programs

Mathematica

Block[{nn = 1600, k = 1, P = 2, a}, a = Most@ Reap[While[Nand[k > 3, Length@ Sow@ Rest@ Select[Range[Min[P, nn]], And[! PrimeQ@ #, CoprimeQ[#, P]] &] == 0], k++; P *= Prime@ k]][[-1, 1]]; Function[b, Map[Count[b, #] &, Union@ b]]@ Flatten@ a] (* Michael De Vlieger, Jun 09 2017 *)

Extensions

Edited by Michael De Vlieger, Jun 09 2017