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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
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A121361 A191907 A052343

Adjacent sequences:  A198065 A198066 A198067 * A198069 A198070 A198071

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Nov 08 2011

STATUS

approved

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Last modified August 2 00:27 EDT 2014. Contains 245137 sequences.