A198067 Square array read by antidiagonals, n>=1, k>=1; T(n,k) is the number of nonprime numbers which are prime to n and are not strong divisors of k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 6, 2, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 7, 1, 6, 2, 3, 1, 3, 1, 2

Offset

1,11

Comments

We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let alpha(n) be number of nonprime numbers in the reduced residue system of n. Then alpha(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({1,4,8,14}) = 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] 1, 1, 1, 1, 1, 1, 1, 1
[2] 1, 1, 1, 1, 1, 1, 1, 1
[3] 1, 1, 1, 1, 1, 1, 1, 1
[4] 1, 1, 1, 1, 1, 1, 1, 1
[5] 2, 2, 2, 1, 2, 2, 2, 1
[6] 1, 1, 1, 1, 1, 1, 1, 1
[7] 3, 3, 3, 2, 3, 2, 3, 2
[8] 1, 1, 1, 1, 1, 1, 1, 1
-
Triangle k=1..n, n>=1:
[1]           1
[2]          1, 1
[3]        1, 1, 1
[4]       1, 1, 1, 1
[5]     2, 2, 2, 1, 2
[6]    1, 1, 1, 1, 1, 1
[7]  3, 3, 3, 2, 3, 2, 3
[8] 1, 1, 1, 1, 1, 1, 1, 1
-
Triangle n=1..k, k>=1:
[1]           1
[2]          1, 1
[3]        1, 1, 1
[4]       1, 1, 1, 1
[5]     1, 1, 1, 1, 2
[6]    1, 1, 1, 1, 2, 1
[7]  1, 1, 1, 1, 2, 1, 3
[8] 1, 1, 1, 1, 1, 1, 2, 1

Maple

strongdivisors := n -> numtheory[divisors](n) minus {1}:
coprimes  := n -> select(k->igcd(k, n)=1, {$1..n}):
nonprimes := n -> remove(isprime, {$1..n});
T := (n, k) -> nops(nonprimes(n) intersect (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, A048864, A193804, A193805, A198066.

Sequence in context: A211111 A074971 A344008 * A282749 A132587 A318930

Adjacent sequences: A198064 A198065 A198066 * A198068 A198069 A198070

Keyword

nonn,tabl

Author

Peter Luschny, Nov 07 2011

Status

approved