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A287918
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Union of nonprime 1 <= t <= m for m in A036913, with gcd(t,m) = 1.
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0
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1, 25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 185, 187, 203, 205, 209, 215, 217, 221, 235, 247, 253, 259, 265, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 391, 395, 403
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OFFSET
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1,2
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COMMENTS
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List of nonprime totatives t of m for m in A036913.
Nonprime 1 is coprime to all numbers, thus a(1) = 1.
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.
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.
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LINKS
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EXAMPLE
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List of nonprime totatives 1 <= t <= m for m <= 210 in A036913:
m: 1 <= t <= m
2: 1;
6: 1;
12: 1;
18: 1;
30: 1;
42: 1, 25;
60: 1, 49;
66: 1, 25, 35, 49, 65;
90: 1, 49, 77;
120: 1, 49, 77, 91, 119;
126: 1, 25, 55, 65, 85, 95, 115, 121, 125;
150: 1, 49, 77, 91, 119, 121, 133, 143;
210: 1, 121, 143, 169, 187, 209;
...
Indices of A036913 of first and last terms m such that gcd(a(n),m)=1:
n a(n) Freq. First Last
-------------------------------
1 1 oo 1 oo
2 25 4 6 18
3 35 1 8 8
4 49 14 7 40
5 55 1 11 11
6 65 3 8 18
7 77 8 9 24
8 85 2 11 18
9 91 11 10 40
10 95 2 11 18
11 115 2 11 18
12 119 9 10 27
13 121 75 11 308
14 125 2 11 18
15 133 10 12 40
16 143 36 12 107
17 145 1 18 18
18 155 1 18 18
19 161 8 14 40
20 169 96 13 248
...
Positions of squared primes q^2 in a(n):
q^2 q
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2 25 5 6 42
4 49 7 7 60
13 121 11 11 126
20 169 13 13 210
35 289 17 16 330
48 361 19 17 420
71 529 23 20 630
107 841 29 25 1050
123 961 31 25 1050
173 1369 37 28 1470
210 1681 41 30 1890
234 1849 43 30 1890
283 2209 47 31 2310
303 2401 49 40 5610
359 2809 53 33 2940
456 3481 59 35 3570
486 3721 61 36 3990
598 4489 67 37 4620
676 5041 71 39 5460
721 5329 73 39 5460
...
(End)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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