A198067 -id:A198067 - cp's OEIS frontend

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

Peter Luschny, Nov 07 2011

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).

			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

Peter Luschny, Euler's totient function

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

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.

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 *)

A198068 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.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 2, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 1, 2, 1, 1, 0, 2, 2, 2, 2, 3, 3, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 0, 1, 2, 2, 2, 2, 2, 1, 2, 2

Offset: 1

Peter Luschny, Nov 08 2011

We say d is a strong divisor of n iff d is a divisor of n and d > 1. Let omega(n) be the number of distinct primes dividing n. Then omega(n) = T(n,1) = T(n,n).

			T(15, 22) = card({2,3,5,11}) = 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] 0, 0, 0, 0, 0, 0, 0, 0
[2] 1, 1, 1, 1, 1, 1, 1, 1
[3] 1, 2, 1, 2, 1, 2, 1, 2
[4] 1, 1, 2, 1, 1, 2, 1, 1
[5] 1, 2, 2, 2, 1, 3, 1, 2
[6] 2, 2, 2, 2, 3, 2, 2, 2
[7] 1, 2, 2, 2, 2, 3, 1, 2
[8] 1, 1, 2, 1, 2, 2, 2, 1
-
Triangle k=1..n, n>=1:
[1]           0
[2]          1, 1
[3]        1, 2, 1
[4]       1, 1, 2, 1
[5]     1, 2, 2, 2, 1
[6]    2, 2, 2, 2, 3, 2
[7]  1, 2, 2, 2, 2, 3, 1
[8] 1, 1, 2, 1, 2, 2, 2, 1
-
Triangle n=1..k, k>=1:
[1]           0
[2]          0, 1
[3]        0, 1, 1
[4]       0, 1, 2, 1
[5]     0, 1, 1, 1, 1
[6]    0, 1, 2, 2, 3, 2
[7]  0, 1, 1, 1, 1, 2, 1
[8] 0, 1, 2, 1, 2, 2, 2, 1

Peter Luschny, Euler's totient function

Cf. A000010, A001221, A193804, A193805, A198066, A198067.

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 ({$1..n} minus (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.

cp's OEIS Frontend

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.

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