cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A193804 Square array read by antidiagonals: S(n,k) = n - A193805(n,k).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 2, 1, 1, 0, 4, 2, 3, 2, 1, 0, 1, 4, 2, 2, 1, 1, 0, 4, 2, 4, 3, 2, 2, 1, 0, 3, 4, 2, 4, 1, 3, 1, 1, 0, 6, 4, 5, 3, 5, 3, 2, 2, 1, 0, 1, 6, 3, 4, 2, 4, 1, 2, 1, 1, 0, 8, 2, 7, 5, 5, 4, 4, 3, 3, 2, 1, 0, 1, 8, 2, 6, 4, 5, 1, 4, 2, 2, 1, 1, 0
Offset: 1

Views

Author

Peter Luschny, Aug 06 2011

Keywords

Comments

Let cophi(n) be the cototient function A051953(n). Then cophi(n) = S(n,1) = S(n,n).

Examples

			[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] 2, 2, 3, 2, 2, 3, 2, 2
[5] 1, 2, 2, 3, 1, 3, 1, 3
[6] 4, 4, 4, 4, 5, 4, 4, 4
[7] 1, 2, 2, 3, 2, 4, 1, 3
[8] 4, 4, 5, 4, 5, 5, 5, 4
Triangle k=1..n, n>=1:
[1]           0
[2]          1, 1
[3]        1, 2, 1
[4]       2, 2, 3, 2
[5]     1, 2, 2, 3, 1
[6]    4, 4, 4, 4, 5, 4
[7]  1, 2, 2, 3, 2, 4, 1
[8] 4, 4, 5, 4, 5, 5, 5, 4
Triangle n=1..k, k>=1:
[1]            0
[2]           0, 1
[3]         0, 1, 1
[4]        0, 1, 2, 2
[5]      0, 1, 1, 2, 1
[6]     0, 1, 2, 3, 3, 4
[7]   0, 1, 1, 2, 1, 4, 1
[8]  0, 1, 2, 2, 3, 4, 3, 4
S(15, 22) = card({2,3,5,6,9,10,11,12,15}) = 9 as
the defining set is {1,2,..,15} minus {1,4,7,8,13,14}.

Links

Crossrefs

Cf. A000010, A051953, A193805.

Programs

  • Maple
    strongdivisors := n -> numtheory[divisors](n) minus {1}:
coprimes := n -> select(k->igcd(k,n)=1,{$1..n}):
S := (n,k) -> nops({seq(i,i={$1..n})}
minus((coprimes(n) minus strongdivisors(k)))):
seq(seq(S(n-k+1,k), k=1..n), n=1..8);  # 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[ Range[n], Complement[ Select[ Range[n], CoprimeQ[#, n]&], Divisors[k] // Rest]] // Length; Table[ s[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

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

Views

Author

Peter Luschny, Nov 07 2011

Keywords

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).

Examples

			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

Links

Crossrefs

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

Programs

  • 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 *)

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

Views

Author

Peter Luschny, Nov 07 2011

Keywords

Comments

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).

Examples

			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

Links

Crossrefs

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

Programs

  • Maple
    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

Views

Author

Peter Luschny, Nov 08 2011

Keywords

Comments

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).

Examples

			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

Links

Crossrefs

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

Programs

  • 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 ({$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.
Showing 1-4 of 4 results.

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