A193804 -id:A193804 - cp's OEIS frontend

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

Peter Luschny, Aug 05 2011

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.

			[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}.

Peter Luschny, Euler's totient function

Cf. A000010, A051953, A193804.

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.

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 *)

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

A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 27, 27, 1, 9, 33, 65, 81, 81, 1, 11, 51, 131, 211, 243, 243, 1, 13, 73, 233, 473, 665, 729, 729, 1, 15, 99, 379, 939, 1611, 2059, 2187, 2187, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 6561, 6561, 1, 19, 163, 835, 2851

Offset: 0

Clark Kimberling, Aug 06 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

			First six rows:
1
1....1
1....3....3
1....5....9....9
1....7....19...27...27
1....9....33...65...81...81

Cf. A193722, A193804. A119258, A193860.

z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193823 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.
O.g.f.: (1 - 2*x*t)^2/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + x)*t + (1 + 3*x + 3*x^2)*t^2 + .... Cf. A193860.
For n >= 1, the n-th row polynomial R(n,x) = 1/(x-1)*( 3^(n-1)*x^(n+1) - (2*x + 1)^(n-1) ). (End)

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.

Original entry on oeis.org

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 *)

A198067 Square array read by antidiagonals, n>=1, k>=1; T(n,k) is the number of nonprime numbers which are prime to n and are not strong divisors of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 6, 2, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 7, 1, 6, 2, 3, 1, 3, 1, 2

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 alpha(n) be number of nonprime numbers in the reduced residue system of n. Then alpha(n) = T(n,1) = T(n,n).

			T(15, 22) = card({1,4,8,14}) = 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] 1, 1, 1, 1, 1, 1, 1, 1
[2] 1, 1, 1, 1, 1, 1, 1, 1
[3] 1, 1, 1, 1, 1, 1, 1, 1
[4] 1, 1, 1, 1, 1, 1, 1, 1
[5] 2, 2, 2, 1, 2, 2, 2, 1
[6] 1, 1, 1, 1, 1, 1, 1, 1
[7] 3, 3, 3, 2, 3, 2, 3, 2
[8] 1, 1, 1, 1, 1, 1, 1, 1
-
Triangle k=1..n, n>=1:
[1]           1
[2]          1, 1
[3]        1, 1, 1
[4]       1, 1, 1, 1
[5]     2, 2, 2, 1, 2
[6]    1, 1, 1, 1, 1, 1
[7]  3, 3, 3, 2, 3, 2, 3
[8] 1, 1, 1, 1, 1, 1, 1, 1
-
Triangle n=1..k, k>=1:
[1]           1
[2]          1, 1
[3]        1, 1, 1
[4]       1, 1, 1, 1
[5]     1, 1, 1, 1, 2
[6]    1, 1, 1, 1, 2, 1
[7]  1, 1, 1, 1, 2, 1, 3
[8] 1, 1, 1, 1, 1, 1, 2, 1

Peter Luschny, Euler's totient function

Cf. A000010, A048864, A193804, A193805, A198066.

strongdivisors := n -> numtheory[divisors](n) minus {1}:
coprimes  := n -> select(k->igcd(k, n)=1, {$1..n}):
nonprimes := n -> remove(isprime, {$1..n});
T := (n,k) -> nops(nonprimes(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.

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

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.

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A193823 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

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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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A198067 Square array read by antidiagonals, n>=1, k>=1; T(n,k) is the number of nonprime numbers which are prime to n and are not strong divisors of k.

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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.

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Maple