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%I #27 Jan 12 2020 10:18:33
%S 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,
%T 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,
%U 6,1,2,3,4,5,7,3,1,4,2,1,6,1,3,2,7,5,4
%N Number of distinct primorials A002110(k) > A285784(n) such that the primorials are coprime to A285784(n).
%C Terms of A285784 that have a(n) = 1 appear in A287390.
%C Terms of A285784 that have a(n) > 1 appear in A287391.
%C From _Michael De Vlieger_, Jun 09 2017: (Start)
%C 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#.
%C 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.
%C 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.
%C 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.
%C 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.
%C 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, ...}
%C 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, ...}
%C (End)
%H Jamie Morken, <a href="/A287917/b287917.txt">Table of n, a(n) for n = 2..10000</a>
%e The sequence starts:
%e n A285784(n) a(n)
%e 2 121 1
%e 3 143 1
%e 4 169 2
%e 5 187 1
%e 6 209 1
%e 7 221 1
%e 8 247 1
%e 9 289 2
%e 10 299 1
%e 11 323 2
%e 12 361 3
%e 13 377 1
%e 14 391 2
%e 15 403 1
%e 16 437 3
%e 17 481 1
%e 18 493 2 ...
%t 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 *)
%Y Cf. A002110, A285784, A287390, A287391.
%K nonn
%O 2,3
%A _Jamie Morken_ and _Michael De Vlieger_, Jun 08 2017
%E Edited by _Michael De Vlieger_, Jun 09 2017
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