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 1-6 of 6 results.

A133233 Triangle A133232 read by rows with an additional column T(n,0)=1 added to the left.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 3, 4, 1, 1, 1, 3, 4, 5, 1, 1, 1, 3, 4, 5, 1, 1, 1, 1, 3, 4, 5, 1, 7, 1, 1, 1, 3, 1, 5, 1, 7, 8, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11
Offset: 0

Views

Author

Mats Granvik, Oct 13 2007

Keywords

Comments

Attaching an additional 1 does not change the composition compared to A133232 since neither the LCM over the elements of a row nor their product is affected.

Examples

			The first rows of the triangle and the least common multiple of the rows are:
lcm{1} = 1
lcm{1, 1} = 1
lcm{1, 1, 2} = 2
lcm{1, 1, 2, 3} = 6
lcm{1, 1, 1, 3, 4} = 12
lcm{1, 1, 1, 3, 4, 5} = 60
lcm{1, 1, 1, 3, 4, 5, 1} = 60
lcm{1, 1, 1, 3, 4, 5, 1, 7} = 420
lcm{1, 1, 1, 3, 1, 5, 1, 7, 8} = 840
lcm{1, 1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520
Multiplying the terms in the rows produces the same result:
1 = 1
1*1 = 1
1*1*2 = 2
1*1*2*3 = 6
1*1*1*3*4 = 12
1*1*1*3*4*5 = 60
1*1*1*3*4*5*1 = 60
1*1*1*3*4*5*1*7 = 420
1*1*1*3*1*5*1*7*8 = 840
1*1*1*1*1*5*1*7*8*9 = 2520

Links

Crossrefs

Cf. A003418, A120112, A000961, A014963.

Formula

T(n,0) = 1.
T(n,k) = A133232(n,k), k>0.

Extensions

Removed information which duplicates A133232; offset set to 0 - R. J. Mathar, Nov 23 2010

A133936 Number of times prime powers occur in the columns of tables A133232 and A133233.

Original entry on oeis.org

0, 2, 6, 4, 20, 0, 42, 8, 18, 0, 110, 0, 156, 0, 0, 16, 272, 0, 342, 0, 0, 0, 506, 0, 100, 0, 54, 0, 812, 0, 930, 32, 0, 0, 0, 0, 1332, 0, 0, 0, 1640, 0, 1806, 0, 0, 0, 2162, 0, 294, 0, 0, 0, 2756, 0, 0, 0, 0, 0, 3422, 0, 3660, 0, 0, 64, 0, 0, 4422, 0, 0, 0, 4970, 0, 5256, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Mats Granvik, Jan 21 2008

Keywords

Crossrefs

Cf. A014963, A100994, A133232, A133233.

Programs

Formula

a(n) = if A014963*A100994A014963*A100994-n

A134579 Column products of tables A133232 and A133233.

Original entry on oeis.org

1, 4, 729, 256, 95367431640625, 0, 311973482284542371301330321821976049, 16777216, 150094635296999121, 0, 3574335935197503226412197580625705154978327969466895714094061686977589739390331653361877238387305580817715435470601
Offset: 1

Views

Author

Mats Granvik, Jan 23 2008

Keywords

Examples

			a(1) = 1^(1*1-1) = 1
a(2) = 2^(2*2-2) = 4
a(3) = 3^(3*3-3) = 729
a(4) = 4^(2*4-4) = 256
a(5) = 5^(5*5-5) = 95367431640625
a(6) = 6^(1*1-6) = 0

Crossrefs

Cf. A133232, A133233, A014963, A100994.

Programs

  • Maple
    A100994 := proc(n) if nops(numtheory[factorset](n)) <> 1 then 1 ; else n ; fi ; end: A014963 := proc(n) if nops(numtheory[factorset](n)) <> 1 then 1 ; else op(1,op(1,ifactors(n)[2])) ; fi ; end: A134579 := proc(n) local e ; e := A014963(n)*A100994(n)-n ; if e >= 0 then n^e ; else 0 ; fi ; end: seq(A134579(n),n=1..13) ; # R. J. Mathar, Jan 30 2008

Formula

a(n) = if A014963(n)*A100994(n)-n >= 0 then n^(A014963(n)*A100994(n)-n) else 0.

Extensions

More terms from R. J. Mathar, Jan 30 2008

A120112 Row sums of number triangle A120111.

Original entry on oeis.org

1, -1, -2, -1, -4, 0, -6, -1, -2, 0, -10, 0, -12, 0, 0, -1, -16, 0, -18, 0, 0, 0, -22, 0, -4, 0, -2, 0, -28, 0, -30, -1, 0, 0, 0, 0, -36, 0, 0, 0, -40, 0, -42, 0, 0, 0, -46, 0, -6, 0, 0, 0, -52, 0, 0, 0, 0, 0, -58, 0, -60, 0, 0, -1, 0, 0, -66, 0, 0, 0, -70, 0, -72
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

The last row of the columns in tables A133232 and A133233 are given by this sequence via the formula: if n < k + k*abs(a(n)) then k, otherwise 1 (1 <= k <= n). - Mats Granvik, Jan 22 2008

Links

Crossrefs

Cf. A014963, A120111, A133232, A133233.

Programs

  • GAP
    Concatenation([1],List([1..70],n->1-Lcm(List([1..n+1],i->i))/Lcm(List([1..n],i->i)))); # Muniru A Asiru, Mar 04 2019
  • Magma
    A120112:= func< n | n eq 0 select 1 else 1-Lcm([1..n+1])/Lcm([1..n]) >;
[A120112(n): n in [0..100]]; // G. C. Greubel, May 05 2023
  • Mathematica
    Table[If[n==0, 1, 1-LCM@@Range[n+1]/(LCM@@Range[n])], {n,0,100}] (* G. C. Greubel, May 05 2023 *)
  • PARI
    a(n) = if (n==0, 1, 1 - lcm(vector(n+1, k, k))/lcm(vector(n, k, k))); \\ Michel Marcus, Sep 11 2016
  • SageMath
    def A120112(n):
    return 1 if n == 0 else 1 - lcm(range(1,n+2)) // lcm(range(1,n+1))
[A120112(n) for n in range(101)] # G. C. Greubel, May 05 2023

Formula

a(n) = 1 - lcm(1,...,n+1)/lcm(1,...,n) for n > 0.
a(n) = 1 - A014963(n+1). - Joerg Arndt, Sep 12 2016

A137152 Triangle read by rows: prime powers whose row products give A051451.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 1, 3, 4, 5, 1, 1, 3, 4, 5, 7, 1, 1, 3, 1, 5, 7, 8, 1, 1, 1, 1, 5, 7, 8, 9, 1, 1, 1, 1, 5, 7, 8, 9, 11, 1, 1, 1, 1, 5, 7, 8, 9, 11, 13, 1, 1, 1, 1, 5, 7, 1, 9, 11, 13, 16, 1, 1, 1, 1, 5, 7, 1, 9, 11, 13, 16, 17, 1, 1, 1, 1, 5, 7, 1, 9, 11, 13, 16, 17, 19, 1, 1, 1, 1, 5, 7, 1
Offset: 1

Views

Author

Mats Granvik, Jan 24 2008

Keywords

Comments

Similar to tables A133232 and A133233.

Examples

			The least common multiple of the first few rows are:
lcm{1} = 1
lcm{1,2} = 2
lcm{1,2,3} = 6
lcm{1,1,3,4} = 12
lcm{1,1,3,4,5} = 60
lcm{1,1,3,4,5,7} = 420
lcm{1,1,3,1,5,7,8} = 840
lcm{1,1,1,1,5,7,8,9} = 2520
lcm{1,1,1,1,5,7,8,9,11} = 27720
Multiplying the terms in the rows produces the same result:
1 = 1
1*2 = 2
1*2*3 = 6
1*1*3*4 = 12
1*1*3*4*5 = 60
1*1*3*4*5*7 = 420
1*1*3*1*5*7*8 = 840
1*1*1*1*5*7*8*9 = 2520
1*1*1*1*5*7*8*9*11 = 27720

Crossrefs

Cf. A051451.

A133255 Triangle with a minimum occurrence of prime powers for which the least common multiple of the rows will give the terms in A003418.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 1, 1, 1, 4, 3, 1, 5, 1, 1, 4, 3, 1, 5, 1, 1, 1, 4, 3, 1, 5, 1, 7, 1, 1, 8, 3, 1, 5, 1, 7, 1, 1, 1, 8, 9, 1, 5, 1, 7, 1, 1, 1, 1, 8, 9, 1, 5, 1, 7, 1, 1, 1, 1, 1, 8, 9, 1, 5, 1, 7, 1, 1, 1, 11, 1, 1, 8, 9, 1, 5, 1, 7, 1, 1, 1, 11, 1, 1, 1, 8, 9, 1, 5, 1, 7, 1, 1, 1, 11, 1
Offset: 1

Views

Author

Mats Granvik, Oct 14 2007

Keywords

Comments

Checked up to 29th row. Similar to A133232 and A133233. In this table the prime powers with the same base are in the same column. A prime power occurs in the table: (base of prime power-1)*(the prime power).

Examples

			lcm{1}= 1
lcm{1,1} = 1
lcm{1,1,2} = 2
lcm{1,1,2,3} = 6
lcm{1,1,4,3,1} = 12
lcm{1,1,4,3,1,5} = 60
lcm{1,1,4,3,1,5,1} = 60
lcm{1,1,4,3,1,5,1,7} = 420
lcm{1,1,8,3,1,5,1,7,1} = 840
lcm{1,1,8,9,1,5,1,7,1,1} = 2520
1 = 1
1*1 = 1
1*1*2 = 2
1*1*2*3 = 6
1*1*4*3*1 = 12
1*1*4*3*1*5 = 60
1*1*4*3*1*5*1 = 60
1*1*4*3*1*5*1*7 = 420
1*1*8*3*1*5*1*7*1 = 840
1*1*8*9*1*5*1*7*1*1 = 2520

Crossrefs

Cf. A089026, A003418, A000961.

Programs

Formula

T(n,k) = if k=1 then 1 elseif n-1>=(A089026(n-1))^0 and n-1<(A089026(n-1))^1 then (A089026(n-1))^0 elseif n-1>=(A089026(n-1))^1 and n-1<(A089026(n-1))^2 then (A089026(n-1))^1 elseif n-1>=(A089026(n-1))^2 and n-1<(A089026(n-1))^3 then (A089026(n-1))^2 elseif n-1>=(A089026(n-1))^3 and n-1<(A089026(n-1))^4 then (A089026(n-1))^3 elseif n-1>=(A089026(n-1))^4 and n-1<(A089026(n-1))^5 then (A089026(n-1))^4 else 1 (1<=k<=n) And so on, this formula needs to be expanded if one wants to make a bigger table. A089026(n-1) means that the index to that sequence is shifted in this formula so that the first term in A089026 is used in the second column of the table.
Showing 1-6 of 6 results.

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