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A287918 Union of nonprime 1 <= t <= m for m in A036913, with gcd(t,m) = 1. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

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)

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 *)

CROSSREFS

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

Sequence in context: A133633 A067793 A038509 * A054550 A107472 A193165

Adjacent sequences:  A287915 A287916 A287917 * A287919 A287920 A287921

KEYWORD

nonn

AUTHOR

Jamie Morken and Michael De Vlieger, Jun 11 2017

STATUS

approved

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Last modified September 15 12:22 EDT 2019. Contains 327078 sequences. (Running on oeis4.)