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

1,4

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.

Links

Table of n, a(n) for n=1..86.

Peter Luschny, Euler's totient function

Example

[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}.

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 *)

Prog

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

Crossrefs

Cf. A000010, A051953, A193804.

Sequence in context: A143258 A027199 A140218 * A159704 A348194 A101428

Adjacent sequences: A193802 A193803 A193804 * A193806 A193807 A193808

Keyword

nonn,nice,tabl

Author

Peter Luschny, Aug 05 2011

Status

approved