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
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}.
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[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
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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
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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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 08 2011
STATUS
approved