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

Author

Michael De Vlieger, Apr 26 2017

Keywords

• nonn
• easy
• tabf

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.

Examples

			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
  ...
		

Links

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

Crossrefs

Cf. A038566, A048597, A048864, A112484.

Programs

Mathematica

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

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