A198068 Square array read by antidiagonals, n>=1, k>=1; T(n,k) is the number of primes which are prime to n and are not strong divisors of k.

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 2, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 1, 2, 1, 1, 0, 2, 2, 2, 2, 3, 3, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 0, 1, 2, 2, 2, 2, 2, 1, 2, 2

Offset

1,8

Comments

We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let omega(n) be the number of distinct primes dividing n. Then omega(n) = T(n,1) = T(n,n).

Links

Table of n, a(n) for n=1..87.

Peter Luschny, Euler's totient function

Example

T(15, 22) = card({2,3,5,11}) = 4 because the coprimes of 15 are {1,2,4,7,8,11,13,14} and the strong divisors of 22 are {2,11,22}.
-
[x][1][2][3][4][5][6][7][8]
[1] 0, 0, 0, 0, 0, 0, 0, 0
[2] 1, 1, 1, 1, 1, 1, 1, 1
[3] 1, 2, 1, 2, 1, 2, 1, 2
[4] 1, 1, 2, 1, 1, 2, 1, 1
[5] 1, 2, 2, 2, 1, 3, 1, 2
[6] 2, 2, 2, 2, 3, 2, 2, 2
[7] 1, 2, 2, 2, 2, 3, 1, 2
[8] 1, 1, 2, 1, 2, 2, 2, 1
-
Triangle k=1..n, n>=1:
[1]           0
[2]          1, 1
[3]        1, 2, 1
[4]       1, 1, 2, 1
[5]     1, 2, 2, 2, 1
[6]    2, 2, 2, 2, 3, 2
[7]  1, 2, 2, 2, 2, 3, 1
[8] 1, 1, 2, 1, 2, 2, 2, 1
-
Triangle n=1..k, k>=1:
[1]           0
[2]          0, 1
[3]        0, 1, 1
[4]       0, 1, 2, 1
[5]     0, 1, 1, 1, 1
[6]    0, 1, 2, 2, 3, 2
[7]  0, 1, 1, 1, 1, 2, 1
[8] 0, 1, 2, 1, 2, 2, 2, 1

Maple

strongdivisors := n -> numtheory[divisors](n) minus {1}:
coprimes  := n -> select(k->igcd(k, n)=1, {$1..n}):
primes := n -> select(isprime, {$1..n});
T := (n, k) -> nops(primes(n) intersect ({$1..n} minus (coprimes(n) minus strongdivisors(k)))):
seq(seq(T(n-k+1, k), k=1..n), n=1..13);  # Square array by antidiagonals.
seq(print(seq(T(n, k), k=1..n)), n=1..8); # Lower triangle.
seq(print(seq(T(n, k), n=1..k)), k=1..8); # Upper triangle.

Crossrefs

Cf. A000010, A001221, A193804, A193805, A198066, A198067.

Sequence in context: A106799 A212210 A127499 * A358194 A121361 A191907

Adjacent sequences: A198065 A198066 A198067 * A198069 A198070 A198071

Keyword

nonn,tabl

Author

Peter Luschny, Nov 08 2011

Status

approved