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 1-7 of 7 results.

A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 8, 8, 32, 144, 432, 2160, 27000, 582120, 7623000, 336936600, 6740402760, 543454231320, 57619849046760, 4683793138766280, 412882704970215480, 88171665744392750520, 12780536107937124847320, 2685589660883755945879560, 942036670625665177379096280
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

Also the Heinz number of row n of A015716 (with zeros removed).
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 multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), so a(6) = prime(1)^4*prime(2)^2 = 144.
The sequence of terms together with their prime indices begins:
               1: {}
               2: {1}
               2: {1}
               8: {1,1,1}
               8: {1,1,1}
              32: {1,1,1,1,1}
             144: {1,1,1,1,2,2}
             432: {1,1,1,1,2,2,2}
            2160: {1,1,1,1,2,2,2,3}
           27000: {1,1,1,2,2,2,3,3,3}
          582120: {1,1,1,2,2,2,3,4,4,5}
         7623000: {1,1,1,2,2,3,3,3,4,5,5}
       336936600: {1,1,1,2,2,3,3,4,5,5,6,7}
      6740402760: {1,1,1,2,2,3,4,4,4,6,6,7,8}
    543454231320: {1,1,1,2,2,3,4,4,5,6,7,8,9,10}
  57619849046760: {1,1,1,2,2,3,4,5,5,6,8,9,10,11,12}

Links

Crossrefs

Cf. A000009, A006128, A015716, A015723, A022629, A056239, A066633, A112798, A246867, A325500, A325504, A325506.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n>(i*(i+1)/2), 0, `if`(n=0, [1, 0], b(n, i-1)+
          (p-> p+[0, p[1]*x^i])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (p-> mul((c-> `if`(c=0, 1, ithprime(c)))(
    coeff(p, x, i)), i=1..degree(p)))(b(n$2)[2]):
seq(a(n), n=0..21);  # Alois P. Heinz, Feb 23 2024
  • Mathematica
    Table[Times@@Prime/@Length/@Split[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]]],{n,0,15}]

Formula

a(n) = A181819(A003963(A325505(n))).
A056239(a(n)) = A015723(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).

A325501 Product of Heinz numbers over all integer partitions of n.

Original entry on oeis.org

1, 2, 12, 240, 120960, 638668800, 15064408719360000, 27259975545259032576000000, 682714624600511148826789083611136000000000, 2948964060660649503322235948384635104494106968064000000000000000
Offset: 0

Views

Author

Gus Wiseman, May 06 2019

Keywords

Comments

Row-products of A215366 (positive integers arranged by sum of prime indices A056239).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The integer partitions of 3 are {(3), (2,1), (1,1,1)}, with Heinz numbers {5,6,8}, with product 240, so a(3) = 240.
The sequence of terms together with their prime indices begins:
          1: {}
          2: {1}
         12: {1,1,2}
        240: {1,1,1,1,2,3}
     120960: {1,1,1,1,1,1,1,2,2,2,3,4}
  638668800: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5}

Links

Crossrefs

Cf. A003963, A006128, A007870, A008284, A056239, A066186, A066633, A112798, A118914, A124010, A145519, A215366, A325500, A325506, A325507.

Programs

  • Mathematica
    Table[Times@@Prime/@(Join@@IntegerPartitions[n]),{n,0,5}]

Formula

A001222(a(n)) = A006128(n).
A056239(a(n)) = A066186(n).
A003963(a(n)) = A007870(n).
A124010(a(n),i) = A066633(n,i).

A325507 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all integer partitions of n.

Original entry on oeis.org

1, 2, 6, 28, 340, 3108, 106932, 2732340, 236790060, 19703562780, 3419598096420, 674127752953380, 264134168649181380, 95825592671995399620, 67662122741507082338220, 50556978553034312461203420, 69259146896604886347745839660, 104191622563656655781003976625020
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

Also the Heinz number of row n of A066633.
The Heinz number of an integer partition or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The integer partitions of 4 are {(4), (3,1), (2,2), (2,1,1), (1,1,1,1)}, with multiset union {1,1,1,1,1,1,1,2,2,2,3,4}, with multiplicities (7,3,1,1), so a(4) = prime(7)*prime(3)*prime(1)*prime(1) = 340.
The sequence of terms together with their prime indices begins:
                        1: {}
                        2: {1}
                        6: {1,2}
                       28: {1,1,4}
                      340: {1,1,3,7}
                     3108: {1,1,2,4,12}
                   106932: {1,1,2,4,8,19}
                  2732340: {1,1,2,3,6,11,30}
                236790060: {1,1,2,3,6,9,19,45}
              19703562780: {1,1,2,3,5,8,15,26,67}
            3419598096420: {1,1,2,3,5,8,13,21,41,97}
          674127752953380: {1,1,2,3,5,7,12,18,31,56,139}
       264134168649181380: {1,1,2,3,5,7,12,17,28,45,83,195}
     95825592671995399620: {1,1,2,3,5,7,11,16,25,38,63,112,272}
  67662122741507082338220: {1,1,2,3,5,7,11,16,24,35,55,87,160,373}

Links

Crossrefs

Cf. A001222, A003963, A006128, A007870, A056239, A066633, A087009, A112798, A215366, A302246, A325500, A325501, A325513.

Programs

  • Mathematica
    Table[Times@@Prime/@Length/@Split[Sort[Join@@IntegerPartitions[n]]],{n,0,15}]

Formula

a(n) = Product_{i = 1..n} prime(A066633(n,i)).
a(n) = A181819(A003963(A325500(n))).
a(n) = A181819(A325501(n)).
A001222(a(n)) = n.
A056239(a(n)) = A006128(n).
For n > 0, A181819(a(n)) = A087009(n + 1).

A325502 Heinz number of row n of Pascal's triangle A007318.

Original entry on oeis.org

2, 4, 12, 100, 2548, 407044, 106023164, 136765353124, 399090759725236, 4445098474836287524, 151287513513627682258436, 12698799587219706700017036196, 3463928752077516667634331415766516, 2591202267595530693505786197581910681796
Offset: 0

Views

Author

Gus Wiseman, May 06 2019

Keywords

Comments

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Every odd-indexed term is a square of a squarefree number.

Examples

			Row n = 5 of Pascal's triangle is (1,5,10,10,5,1), with Heinz number prime(1)*prime(5)*prime(10)*prime(10)*prime(5)*prime(1) = 407044, so a(5) = 407044.
The sequence of terms together with their prime indices begins:
                    2: {1}
                    4: {1,1}
                   12: {1,1,2}
                  100: {1,1,3,3}
                 2548: {1,1,4,4,6}
               407044: {1,1,5,5,10,10}
            106023164: {1,1,6,6,15,15,20}
         136765353124: {1,1,7,7,21,21,35,35}
      399090759725236: {1,1,8,8,28,28,56,56,70}
  4445098474836287524: {1,1,9,9,36,36,84,84,126,126}

Links

Crossrefs

Cf. A000040, A001222, A001405, A007318, A056239, A112798, A145519, A215366, A325500, A325503, A325505, A325514.

Programs

  • Mathematica
    Times@@@Table[Prime[Binomial[n,k]],{n,0,5},{k,0,n}]

Formula

A061395(a(n)) = A001405(n).
A056239(a(n)) = A000079(n).
A181819(a(n)) = A038754(n + 1).

A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

Original entry on oeis.org

2, 4, 20, 884, 528844, 3460086044, 340672148731996, 477782556719729075524, 11694209380474301218263758996, 4967476846044415922850025924897606724, 43298471669920632729336800855543564573041217668, 7790810575556906457316064931238939360882160372451591124244
Offset: 1

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}
                              4: {1,1}
                             20: {1,1,3}
                            884: {1,1,6,7}
                         528844: {1,1,10,15,25}
                     3460086044: {1,1,15,31,65,90}
                340672148731996: {1,1,21,63,140,301,350}
          477782556719729075524: {1,1,28,127,266,966,1050,1701}
  11694209380474301218263758996: {1,1,36,255,462,2646,3025,6951,7770}

Links

Crossrefs

Cf. A000040, A001222, A002870, A008277, A024412, A056239, A112798, A215366, A325500, A325502.

Programs

  • Mathematica
    Times@@@Table[Prime[StirlingS2[n,k]],{n,1,10},{k,1,n}]

Formula

a(n) = Product_{i = 1..n} prime(A008277(n,i)).
A061395(a(n)) = A002870(n).
A056239(a(n)) = A000110(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 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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