A285788 Irregular triangle T(n,m): nonprime 1 <= k <= n such that n and k are coprime.

1, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 1, 4, 8, 1, 9, 1, 4, 6, 8, 9, 10, 1, 1, 4, 6, 8, 9, 10, 12, 1, 9, 1, 4, 8, 14, 1, 9, 15, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 1, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 1, 9, 1, 4, 8, 10, 16, 20, 1, 9, 15, 21, 1, 4, 6, 8, 9, 10

Offset

1,6

Comments

Row n is a subset of A038566(n) such that the union of a(n) and A112484(n) = A038566(n).

Row lengths are A048864(n) = A000010(n)-(A000720(n)-A001221(n)), i.e., phi(n)-(pi(n)-omega(n)).

1 appears in every row since 1 is not prime and coprime to all n.

4 is the smallest composite and appears first in row 5 since 4 divides 4.

Rows that contain the single term 1 are in A048597; the largest n = 30 such that the only term is 1.

For prime p, row p contains 1 and all composites k < p, since 1 < m < p are coprime to p.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11055 (rows 1 <= n <= 240)

Example

Triangle begins:
  n\m  1  2   3   4  5   6   7
   1:  1
   2:  1
   3:  1
   4:  1
   5:  1  4
   6:  1
   7:  1  4   6
   8:  1
   9:  1  4   8
  10:  1  9
  11:  1  4   6   8  9  10
  12:  1
  13:  1  4   6   8  9  10  12
  14:  1  9
  15:  1  4   8  14
  16:  1  9  15
  ...

Mathematica

Table[Select[Range@ n, And[! PrimeQ@ #, CoprimeQ[#, n]] &], {n, 23}] // Flatten

Prog

(Python)
from sympy import gcd, isprime
def a(n): return list(filter(lambda k: isprime(k)==0 and gcd(k, n)==1, range(1, n + 1)))
for n in range(1, 21): print a(n) # Indranil Ghosh, Apr 26 2017

Crossrefs

Cf. A038566, A048597, A048864, A112484.

Sequence in context: A222360 A222371 A222479 * A293434 A091570 A116669

Adjacent sequences: A285785 A285786 A285787 * A285789 A285790 A285791

Keyword

nonn,easy,tabf

Author

Michael De Vlieger, Apr 26 2017

Status

approved