login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A101428 A307223

Adjacent sequences:  A193802 A193803 A193804 * A193806 A193807 A193808

KEYWORD

nonn,nice,tabl

AUTHOR

Peter Luschny, Aug 05 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)