A193805 Square array read by antidiagonals: S(n,k) is the number of m which are prime to n and are not strong divisors of k.

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 6, 2, 3, 2, 2, 1, 1, 4, 5, 2, 2, 2, 1, 1, 1, 6, 4, 5, 2, 4, 1, 2, 1, 1, 4, 5, 3, 4, 1, 2, 2, 1, 1, 1, 10, 4, 6, 4, 5, 2, 4, 2, 2, 1, 1, 4, 9, 3, 4, 3, 3, 2, 2, 1, 1, 1, 1, 12, 4, 9, 4, 5, 3, 6, 2

Offset: 1

Author

Peter Luschny, Aug 05 2011

Keywords

• nonn
• nice
• tabl

Comments

We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let phi(n) be Euler's totient function. Then phi(n) = S(n,1) = S(n,n). Thus S(n,k) can be regarded as a generalization of the totient function.

Examples

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

Links

• Peter Luschny, Euler's totient function

Crossrefs

Cf. A000010, A051953, A193804.

Programs

Maple

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

Mathematica

s[n_, k_] := Complement[ Select[ Range[n], GCD [#, n] == 1 &], Rest[ Divisors[k]]] // Length; Table[ s[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2013 *)

PARI

S(n,k)=eulerphi(n)-sumdiv(k,d, gcd(d,n)==1 && d1)
for(s=2,15, for(k=1,s-1, print1(S(s-k,k)", "))) \\ Charles R Greathouse IV, Aug 01 2016