A198066 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, 0, 1, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 3, 1, 1, 1, 1, 0, 0, 3, 2, 1, 1, 1, 0, 0, 0, 3, 3, 2, 1, 2, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 1, 0, 0, 0, 4, 2, 3, 3, 2, 1, 2, 1, 1, 0, 0, 3, 3, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 5, 3, 3, 2, 2, 2, 3, 1, 1, 1, 1, 0, 0

Offset

1,11

Comments

We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let prime_phi(n) be number of primes in the reduced residue system mod n. Then prime_phi(n) = T(n,1) = T(n,n).

Links

Table of n, a(n) for n=1..91.

Peter Luschny, Euler's totient function

Example

T(15, 22) = card({7,13}) = 2 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] 0, 0, 0, 0, 0, 0, 0, 0
[3] 1, 0, 1, 0, 1, 0, 1, 0
[4] 1, 1, 0, 1, 1, 0, 1, 1
[5] 2, 1, 1, 1, 2, 0, 2, 1
[6] 1, 1, 1, 1, 0, 1, 1, 1
[7] 3, 2, 2, 2, 2, 1, 3, 2
[8] 3, 3, 2, 3, 2, 2, 2, 3
-
Triangle k=1..n, n>=1:
[1]           0
[2]          0, 0
[3]        1, 0, 1
[4]       1, 1, 0, 1
[5]     2, 1, 1, 1, 2
[6]    1, 1, 1, 1, 0, 1
[7]  3, 2, 2, 2, 2, 1, 3
[8] 3, 3, 2, 3, 2, 2, 2, 3
-
Triangle n=1..k, k>=1:
[1]            0
[2]           0, 0
[3]         0, 0, 1
[4]        0, 0, 0, 1
[5]      0, 0, 1, 1, 2
[6]     0, 0, 0, 0, 0, 1
[7]   0, 0, 1, 1, 2, 1, 3
[8]  0, 0, 0, 1, 1, 1, 2, 3

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 (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.

Mathematica

T[n_, k_] := Complement[Select[Range[n-1], PrimeQ[#] && CoprimeQ[#, n]&], Rest[Divisors[k]]] // Length;
Table[T[n-k+1, k], {n, 1, 13}, {k, 1, n}] (* Jean-François Alcover, Jun 29 2019 *)

Crossrefs

Cf. A000010, A048865, A193804, A193805, A198067.

Sequence in context: A004610 A068934 A035200 * A141664 A056979 A087812

Adjacent sequences: A198063 A198064 A198065 * A198067 A198068 A198069

Keyword

nonn,tabl

Author

Peter Luschny, Nov 07 2011

Status

approved