OFFSET
2,3
COMMENTS
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)
LINKS
Jamie Morken, Table of n, a(n) for n = 2..10000
EXAMPLE
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 ...
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jamie Morken and Michael De Vlieger, Jun 08 2017
EXTENSIONS
Edited by Michael De Vlieger, Jun 09 2017
STATUS
approved