A203417 -id:A203417

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A203415

Vandermonde determinant of the first n nonprimes (A018252).

+10
5

1, 3, 30, 1680, 201600, 87091200, 1103619686400, 275463473725440000, 240529195987579699200000, 1163776461866305616609280000000, 344605941225348705438623229542400000000, 3717059729911125118574880410324812431360000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Each term divides its successor, as in A203416, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203417.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..42

MATHEMATICA

z=20;

nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)

f[j_]:= nonprime[[j]];

v[n_]:= Product[Product[f[k] - f[j], {j, 1, k-1}], {k, 2, n}];

d[n_]:= Product[(i-1)!, {i, 1, n}];

Table[v[n], {n, 1, z}] (* this sequence *)

Table[v[n+1]/v[n], {n, 1, z}] (* A203416 *)

Table[v[n]/d[n], {n, 1, z}] (* A203417 *)

PROG

(Magma)

A018252:=[n : n in [1..250] | not IsPrime(n) ];

A203415:= func< n | n eq 1 select 1 else (&*[(&*[A018252[k+2] - A018252[j+1]: j in [0..k]]): k in [0..n-2]]) >;

[A203415(n): n in [1..30]]; // G. C. Greubel, Feb 29 2024

(SageMath)

A018252=[n for n in (1..250) if not is_prime(n)]

def A203415(n): return product(product(A018252[k+1]-A018252[j] for j in range(k+1)) for k in range(n-1))

[A203415(n) for n in range(1, 31)] # G. C. Greubel, Feb 29 2024

CROSSREFS

Cf. A000040, A018252, A203416, A203417.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 01 2012

STATUS

approved

A203416

a(n) = A203415(n+1)/A203415(n).

+10
4

3, 10, 56, 120, 432, 12672, 249600, 873180, 4838400, 296110080, 10786406400, 49621572000, 355053404160, 34613526528000, 211189410432000, 1910897049600000, 21311651380219200, 274774815041126400, 62908970812047360000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..350

MATHEMATICA

z=20;

nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)

f[j_]:= nonprime[[j]];

v[n_]:= Product[Product[f[k] - f[j], {j, 1, k-1}], {k, 2, n}];

d[n_]:= Product[(i-1)!, {i, 1, n}];

Table[v[n], {n, 1, z}] (* A203415 *)

Table[v[n+1]/v[n], {n, 1, z}] (* this sequence *)

Table[v[n]/d[n], {n, 1, z}] (* A203417 *)

PROG

(Magma)

A018252:=[n : n in [1..250] | not IsPrime(n) ];

A203416:= func< n | n eq 1 select 3 else (&*[A018252[n+1] - A018252[j+1]: j in [0..n-1]]) >;

[A203416(n): n in [1..30]]; // G. C. Greubel, Feb 29 2024

(SageMath)

A018252=[n for n in (1..250) if not is_prime(n)]

def A203416(n): return product(A018252[n]-A018252[j] for j in range(n))

[A203416(n) for n in range(1, 31)] # G. C. Greubel, Feb 29 2024

CROSSREFS

Cf. A000040, A018252, A203415, A203417.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 01 2012

STATUS

approved

A203420

a(n) = A203418(n)/A000178(n).

+10
4

1, 2, 8, 20, 40, 384, 10240, 126720, 1013760, 48660480, 7612661760, 473174507520, 16701626253312, 4036421002199040, 407426244909465600, 23814785343474892800, 932976775107465707520, 26694111965427724713984, 9044593230639040844267520

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..140

R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

MATHEMATICA

composite = Select[Range[100], CompositeQ]; (* A002808 *)

z = 20;

f[j_]:= composite[[j]];

v[n_]:= Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}];

d[n_]:= Product[(i-1)!, {i, 1, n}];

Table[v[n], {n, z}] (* A203418 *)

Table[v[n+1]/v[n], {n, z}] (* A203419 *)

Table[v[n]/d[n], {n, z}] (* this sequence *)

PROG

(Magma)

A002808:=[n: n in [2..250] | not IsPrime(n)];

BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;

a:= func< n | n eq 1 select 1 else (&*[(&*[A002808[k+2] - A002808[j+1]: j in [0..k]]): k in [0..n-2]])/BarnesG(n+1) >;

[a(n): n in [1..40]]; // G. C. Greubel, Feb 24 2024

(SageMath)

A002808=[n for n in (2..250) if not is_prime(n)]

def BarnesG(n): return product(factorial(j) for j in range(1, n-1))

def a(n): return product(product(A002808[k+1] - A002808[j] for j in range(k+1)) for k in range(n-1))/BarnesG(n+1)

[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 24 2024

CROSSREFS

Cf. A000040, A000178, A002808, A018252, A202808, A203417, A203418, A203419.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 02 2012

STATUS

approved

A203529

v(n)/A000178(n); v=A203527 and A000178=(superfactorials).

+10
3

1, 5, 175, 44100, 60913125, 427756329000, 20552836095792000, 6952965728817588480000, 11895516181976215338950400000, 99606443887767729350960121600000000, 5830034964946921746536425070101217280000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

It is conjectured that every term of A203529 is an integer.

LINKS

Table of n, a(n) for n=1..11.

MATHEMATICA

t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];

nonprime = Rest[Union[t]] (* A018252 *)

f[j_] := nonprime[[j]]; z = 20;

v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]

d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)

Table[v[n], {n, 1, z}] (* A203527 *)

Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A203528 *)

Table[v[n]/d[n], {n, 1, 20}] (* A203529 *)

CROSSREFS

Cf. A203527, A203528, A018252, A093883, A203417.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 03 2012

STATUS

approved

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