A287918 Union of nonprime 1 <= t <= m for m in A036913, with gcd(t,m) = 1.

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

Offset: 1

Author

Jamie Morken and Michael De Vlieger, Jun 11 2017

Keywords

• nonn

Comments

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.

Examples

			From _Michael De Vlieger_, Jun 14 2017: (Start)
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
    n   a(n)  sqrt(a(n))     k    m = A036913(k)
  ----------------------------------------------
    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)
		

Crossrefs

Cf. A001248, A036913, A038509, A067793, A285784, A287917.

Programs

Mathematica

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 *)