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%I #13 Sep 24 2018 02:40:59
%S 1,25,35,49,55,65,77,85,91,95,115,119,121,125,133,143,145,155,161,169,
%T 185,187,203,205,209,215,217,221,235,247,253,259,265,287,289,295,299,
%U 301,305,319,323,325,329,335,341,343,355,361,365,371,377,391,395,403
%N Union of nonprime 1 <= t <= m for m in A036913, with gcd(t,m) = 1.
%C List of nonprime totatives t of m for m in A036913.
%C Nonprime 1 is coprime to all numbers, thus a(1) = 1.
%C The integers {175, 245, 275} are absent, distinguishing this sequence from A038509 and A067793. These terms have factors 5^2 * 7, 5 * 7^2, 5^2 * 11. Only the terms in positions {2, 3, 4, 6, 8, 11, 18} of A036913 (i.e., {6, 12, 18, 42, 66, 126, 462}) are larger and coprime to 5. Of these only 462 is greater than these three terms, however 462 is divisible by 7 and 11. Thus {175, 245, 275} are not terms.
%C Squared primes q^2 for q >= 5 appear in the sequence at positions {2, 4, 13, 20, 35, 48, 71, 107, 123, 173, ...}. These are coprime to and smaller than {42, 60, 126, 210, 330, 420, ...} at indices {6, 7, 11, 13, 16, 17, 20, 25, 25, 28, 30, 30, 31, 40, 33, 35, ...} in A036913.
%e From _Michael De Vlieger_, Jun 14 2017: (Start)
%e List of nonprime totatives 1 <= t <= m for m <= 210 in A036913:
%e m: 1 <= t <= m
%e 2: 1;
%e 6: 1;
%e 12: 1;
%e 18: 1;
%e 30: 1;
%e 42: 1, 25;
%e 60: 1, 49;
%e 66: 1, 25, 35, 49, 65;
%e 90: 1, 49, 77;
%e 120: 1, 49, 77, 91, 119;
%e 126: 1, 25, 55, 65, 85, 95, 115, 121, 125;
%e 150: 1, 49, 77, 91, 119, 121, 133, 143;
%e 210: 1, 121, 143, 169, 187, 209;
%e ...
%e Indices of A036913 of first and last terms m such that gcd(a(n),m)=1:
%e n a(n) Freq. First Last
%e -------------------------------
%e 1 1 oo 1 oo
%e 2 25 4 6 18
%e 3 35 1 8 8
%e 4 49 14 7 40
%e 5 55 1 11 11
%e 6 65 3 8 18
%e 7 77 8 9 24
%e 8 85 2 11 18
%e 9 91 11 10 40
%e 10 95 2 11 18
%e 11 115 2 11 18
%e 12 119 9 10 27
%e 13 121 75 11 308
%e 14 125 2 11 18
%e 15 133 10 12 40
%e 16 143 36 12 107
%e 17 145 1 18 18
%e 18 155 1 18 18
%e 19 161 8 14 40
%e 20 169 96 13 248
%e ...
%e Positions of squared primes q^2 in a(n):
%e q^2 q
%e n a(n) sqrt(a(n)) k m = A036913(k)
%e ----------------------------------------------
%e 2 25 5 6 42
%e 4 49 7 7 60
%e 13 121 11 11 126
%e 20 169 13 13 210
%e 35 289 17 16 330
%e 48 361 19 17 420
%e 71 529 23 20 630
%e 107 841 29 25 1050
%e 123 961 31 25 1050
%e 173 1369 37 28 1470
%e 210 1681 41 30 1890
%e 234 1849 43 30 1890
%e 283 2209 47 31 2310
%e 303 2401 49 40 5610
%e 359 2809 53 33 2940
%e 456 3481 59 35 3570
%e 486 3721 61 36 3990
%e 598 4489 67 37 4620
%e 676 5041 71 39 5460
%e 721 5329 73 39 5460
%e ...
%e (End)
%t With[{nn = 403, s = Union@FoldList[Max, Values[#][[All, -1]]] &@ KeySort@ PositionIndex@ EulerPhi@ Range[Product[Prime@ i, {i, 8}]]}, Union@ Flatten@ Map[Function[n, Select[Range@ Min[n, nn], And[CoprimeQ[#, n], ! PrimeQ@ #] &]], s]] (* _Michael De Vlieger_, Jun 14 2017 *)
%Y Cf. A001248, A036913, A038509, A067793, A285784, A287917.
%K nonn
%O 1,2
%A _Jamie Morken_ and _Michael De Vlieger_, Jun 11 2017