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.

A325504 Product of products of parts over all strict integer partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 120, 1440, 40320, 1209600, 1567641600, 2633637888000, 13905608048640000, 5046067048690483200000, 5289893008483207348224000000, 1266933607446134946465526579200000000, 99304891373531545064656621572980736000000000000
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Examples

			The strict partitions of 5 are {(5), (4,1), (3,2)} with product a(5) = 5*4*1*3*2 = 120.
The sequence of terms together with their prime indices begins:
              1: {}
              1: {}
              2: {1}
              6: {1,2}
             12: {1,1,2}
            120: {1,1,1,2,3}
           1440: {1,1,1,1,1,2,2,3}
          40320: {1,1,1,1,1,1,1,2,2,3,4}
        1209600: {1,1,1,1,1,1,1,1,2,2,2,3,3,4}
     1567641600: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4}
  2633637888000: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4}

Links

Crossrefs

Cf. A000009, A006128, A007870 (non-strict version), A015723, A022629 (sum of products of parts), A066186, A066189, A066633, A246867, A325505, A325506, A325512, A325513, A325515.

Programs

  • Maple
    a:= n-> mul(i, i=map(x-> x[], select(x->
        nops(x)=nops({x[]}), combinat[partition](n)))):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0, 1], ((f, g)->
     [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
  • Mathematica
    Table[Times@@Join@@Select[IntegerPartitions[n],UnsameQ@@#&],{n,0,10}]

Formula

A001222(a(n)) = A325515(n).
a(n) = A003963(A325506(n)).

A325506 Product of Heinz numbers over all strict integer partitions of n.

Original entry on oeis.org

1, 2, 3, 30, 70, 2310, 180180, 21441420, 6401795400, 200984366583000, 41615822944675980000, 10515527757483671302380000, 4919824049783476260137727416400000, 5158181210492841550866520676965246284000000, 29776760895364738730693151196801613158042403043600000000
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

a(n) is the product of row n of A246867 (squarefree numbers arranged by sum of prime indices).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with Heinz numbers {13,22,21,30}, with product 13*22*21*30 = 180180, so a(6) = 180180.
The sequence of terms together with their prime indices begins:
                     1: {}
                     2: {1}
                     3: {2}
                    30: {1,2,3}
                    70: {1,3,4}
                  2310: {1,2,3,4,5}
                180180: {1,1,2,2,3,4,5,6}
              21441420: {1,1,2,2,3,4,4,5,6,7}
            6401795400: {1,1,1,2,2,3,3,4,5,5,6,7,8}
       200984366583000: {1,1,1,2,2,2,3,3,3,4,4,5,5,6,6,7,8,9}
  41615822944675980000: {1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,6,6,7,7,8,9,10}

Crossrefs

Cf. A003963, A006128, A015723, A022629, A056239, A112798, A147655, A215366, A246867, A325501, A325504, A325505, A325512, A325513.

Programs

  • Mathematica
    Table[Times@@Prime/@(Join@@Select[IntegerPartitions[n],UnsameQ@@#&]),{n,0,15}]

Formula

a(n) = Product_{i = 1..A000009(n)} A246867(n,i).
A001222(a(n)) = A015723(n).
A056239(a(n)) = A066189(n).
A003963(a(n)) = A325504(n).
a(n) = A003963(A325505(n)).

A325505 Heinz number of the set of Heinz numbers of all strict integer partitions of n.

Original entry on oeis.org

2, 3, 5, 143, 493, 62651, 26718511, 22017033127, 44220524211551, 52289759420183033963, 546407750301194131199484983, 8362548333129019658779663581495109, 1828111016191440393570169991636207115709029581, 1059934964500839879758659437301868941873808925011368355891
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

The Heinz number of a set or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also Heinz numbers of rows of A246867 (squarefree numbers arranged by sum of prime indices A056239).

Examples

			The strict integer partitions of 5 are {(5), (4,1), (3,2)}, with Heinz numbers {11,14,15}, with Heinz number prime(11)*prime(14)*prime(15) = 62651, so a(6) = 62651.
The sequence of terms together with their prime indices begins:
                            2: {1}
                            3: {2}
                            5: {3}
                          143: {5,6}
                          493: {7,10}
                        62651: {11,14,15}
                     26718511: {13,21,22,30}
                  22017033127: {17,26,33,35,42}
               44220524211551: {19,34,39,55,66,70}
         52289759420183033963: {23,38,51,65,77,78,105,110}
  546407750301194131199484983: {29,46,57,85,91,102,130,154,165,210}

Links

Crossrefs

Cf. A001222, A003963, A015723, A056239, A066189, A112798, A145519, A147655, A215366, A246867, A325500 (non-strict version), A325504, A325506, A325512, A325513.

Programs

  • Mathematica
    Table[Times@@Prime/@(Times@@Prime/@#&/@Select[IntegerPartitions[n],UnsameQ@@#&]),{n,7}]

Formula

a(n) = Product_{i = 1..A000009(n)} prime(A246867(n,i)).
A001221(a(n)) = A001222(a(n)) = A000009(n).
A056239(a(n)) = A147655(n).
A003963(a(n)) = A325506(n).

A325514 Heinz number of row n of the triangle of partition numbers A008284.

Original entry on oeis.org

2, 2, 4, 8, 24, 72, 600, 4200, 101640, 2042040, 107869080, 6435365640, 644779672680, 62219208188280, 14408598135902520, 3195700205016233640, 1246437353286578234760, 527744165981695537415640, 417665868515500206974318760, 314096677106179199154141208440
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
             2: {1}
             2: {1}
             4: {1,1}
             8: {1,1,1}
            24: {1,1,1,2}
            72: {1,1,1,2,2}
           600: {1,1,1,2,3,3}
          4200: {1,1,1,2,3,3,4}
        101640: {1,1,1,2,3,4,5,5}
       2042040: {1,1,1,2,3,4,5,6,7}
     107869080: {1,1,1,2,3,5,5,7,8,9}
    6435365640: {1,1,1,2,3,5,5,7,10,10,11}
  644779672680: {1,1,1,2,3,5,6,7,11,12,13,15}

Links

Crossrefs

Cf. A000041, A002569, A008284, A056239, A112798, A145519, A215366, A325500, A325502, A325503, A325512, A325514.

Programs

  • Mathematica
    Times@@@Table[If[n>0&&k==0,1,Prime[Length[IntegerPartitions[n,{k}]]]],{n,0,20},{k,0,n}]

Formula

A001221(a(n)) = A325512(n).
A061395(a(n)) = A002569(n).
A056239(a(n)) = A000041(n).
Showing 1-4 of 4 results.

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