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.

Previous Showing 21-26 of 26 results.

A363237 Number of partitions of n with rank a multiple of 5.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 4, 6, 8, 12, 15, 21, 27, 34, 47, 59, 77, 98, 125, 160, 200, 251, 315, 390, 488, 602, 744, 913, 1120, 1370, 1669, 2029, 2462, 2975, 3597, 4327, 5203, 6237, 7466, 8919, 10634, 12653, 15035, 17824, 21114, 24950, 29455, 34705, 40844, 47991, 56317, 65987, 77231, 90252
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363233, A363238, A363239.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 5)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(5*k))/(1-x^(5*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(5*k)) / (1-x^(5*k)).

A363238 Number of partitions of n with rank a multiple of 6.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 5, 2, 6, 6, 10, 11, 21, 19, 32, 37, 51, 59, 90, 97, 138, 162, 215, 253, 340, 392, 514, 610, 771, 916, 1166, 1367, 1711, 2032, 2503, 2965, 3647, 4293, 5237, 6188, 7469, 8808, 10613, 12459, 14920, 17530, 20862, 24457, 29029, 33924, 40099, 46829, 55101, 64215, 75386
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363233, A363237, A363239.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 6)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(6*k))/(1-x^(6*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(6*k)) / (1-x^(6*k)).

A363239 Number of partitions of n with rank a multiple of 7.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 4, 4, 6, 8, 11, 15, 19, 26, 33, 43, 55, 70, 89, 114, 144, 179, 225, 280, 348, 430, 532, 653, 800, 978, 1193, 1449, 1758, 2127, 2569, 3091, 3717, 4455, 5334, 6369, 7596, 9039, 10739, 12734, 15080, 17822, 21039, 24791, 29176, 34277, 40227, 47133, 55165, 64468
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Crossrefs

Cf. A000041, A328988, A340601, A363233, A363237, A363238.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 7)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55);  # Alois P. Heinz, May 23 2023
  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(7*k))/(1-x^(7*k))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(7*k)) / (1-x^(7*k)).

A340929 Heinz numbers of integer partitions of odd negative rank.

Original entry on oeis.org

4, 12, 16, 18, 27, 40, 48, 60, 64, 72, 90, 100, 108, 112, 135, 150, 160, 162, 168, 192, 225, 240, 243, 250, 252, 256, 280, 288, 352, 360, 375, 378, 392, 400, 420, 432, 448, 528, 540, 567, 588, 600, 625, 630, 640, 648, 672, 700, 768, 792, 810, 832, 880, 882
Offset: 1

Views

Author

Gus Wiseman, Jan 29 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The sequence of partitions together with their Heinz numbers begins:
       4: (1,1)             150: (3,3,2,1)
      12: (2,1,1)           160: (3,1,1,1,1,1)
      16: (1,1,1,1)         162: (2,2,2,2,1)
      18: (2,2,1)           168: (4,2,1,1,1)
      27: (2,2,2)           192: (2,1,1,1,1,1,1)
      40: (3,1,1,1)         225: (3,3,2,2)
      48: (2,1,1,1,1)       240: (3,2,1,1,1,1)
      60: (3,2,1,1)         243: (2,2,2,2,2)
      64: (1,1,1,1,1,1)     250: (3,3,3,1)
      72: (2,2,1,1,1)       252: (4,2,2,1,1)
      90: (3,2,2,1)         256: (1,1,1,1,1,1,1,1)
     100: (3,3,1,1)         280: (4,3,1,1,1)
     108: (2,2,2,1,1)       288: (2,2,1,1,1,1,1)
     112: (4,1,1,1,1)       352: (5,1,1,1,1,1)
     135: (3,2,2,2)         360: (3,2,2,1,1,1)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101707.
The positive version is A101707 (A340604).
The even version is A101708 (A340930).
The not necessarily odd version is A064173 (A340788).
A001222 counts prime factors.
A027193 counts partitions of odd length (A026424).
A047993 counts balanced partitions (A106529).
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank equal to maximum minus minimum part (A324515).
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.

Programs

  • Mathematica
    rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&]

Formula

For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.

A340930 Heinz numbers of integer partitions of even negative rank.

Original entry on oeis.org

8, 24, 32, 36, 54, 80, 81, 96, 120, 128, 144, 180, 200, 216, 224, 270, 300, 320, 324, 336, 384, 405, 450, 480, 486, 500, 504, 512, 560, 576, 675, 704, 720, 729, 750, 756, 784, 800, 840, 864, 896, 1056, 1080, 1125, 1134, 1176, 1200, 1250, 1260, 1280, 1296, 1344
Offset: 1

Views

Author

Gus Wiseman, Jan 30 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The sequence of partitions together with their Heinz numbers begins:
       8: (1,1,1)             270: (3,2,2,2,1)
      24: (2,1,1,1)           300: (3,3,2,1,1)
      32: (1,1,1,1,1)         320: (3,1,1,1,1,1,1)
      36: (2,2,1,1)           324: (2,2,2,2,1,1)
      54: (2,2,2,1)           336: (4,2,1,1,1,1)
      80: (3,1,1,1,1)         384: (2,1,1,1,1,1,1,1)
      81: (2,2,2,2)           405: (3,2,2,2,2)
      96: (2,1,1,1,1,1)       450: (3,3,2,2,1)
     120: (3,2,1,1,1)         480: (3,2,1,1,1,1,1)
     128: (1,1,1,1,1,1,1)     486: (2,2,2,2,2,1)
     144: (2,2,1,1,1,1)       500: (3,3,3,1,1)
     180: (3,2,2,1,1)         504: (4,2,2,1,1,1)
     200: (3,3,1,1,1)         512: (1,1,1,1,1,1,1,1,1)
     216: (2,2,2,1,1,1)       560: (4,3,1,1,1,1)
     224: (4,1,1,1,1,1)       576: (2,2,1,1,1,1,1,1)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101708.
The positive version is (A340605).
The odd version is A101707 (A340929).
The not necessarily even version is A064173 (A340788).
A001222 counts prime factors.
A027187 counts partitions of even length.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058696 counts partitions of even numbers.
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A006141, A039900, A117193, A117409, A316413, A324517, A325134, A326845, A340604, A340787, A340828, A340830.

Programs

  • Mathematica
    rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],EvenQ[rk[#]]&&rk[#]<0&]

A341448 Heinz numbers of integer partitions of type OO.

Original entry on oeis.org

6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240
Offset: 1

Views

Author

Gus Wiseman, Feb 15 2021

Keywords

Comments

These partitions are defined to have an odd number of odd parts and an odd number of even parts. They also have even length and odd sum.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      6: (2,1)         74: (12,1)           141: (15,2)
     14: (4,1)         77: (5,4)            142: (20,1)
     15: (3,2)         86: (14,1)           143: (6,5)
     24: (2,1,1,1)     93: (11,2)           145: (10,3)
     26: (6,1)         95: (8,3)            150: (3,3,2,1)
     33: (5,2)         96: (2,1,1,1,1,1)    152: (8,1,1,1)
     35: (4,3)        104: (6,1,1,1)        158: (22,1)
     38: (8,1)        106: (16,1)           161: (9,4)
     51: (7,2)        119: (7,4)            177: (17,2)
     54: (2,2,2,1)    122: (18,1)           178: (24,1)
     56: (4,1,1,1)    123: (13,2)           185: (12,3)
     58: (10,1)       126: (4,2,2,1)        201: (19,2)
     60: (3,2,1,1)    132: (5,2,1,1)        202: (26,1)
     65: (6,3)        135: (3,2,2,2)        204: (7,2,1,1)
     69: (9,2)        140: (4,3,1,1)        209: (8,5)

Crossrefs

Note: A-numbers of ranking sequences are in parentheses below.
The case of odd parts, length, and sum is counted by A078408 (A300272).
The type EE version is A236913 (A340784).
These partitions (for odd n) are counted by A236914.
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd (A340932).
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A160786 counts odd-length partitions of odd numbers (A340931).
A340101 counts factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A000700, A024429, A027187, A106529, A117409, A174725, A257541, A325134, A339890, A340102, A340604.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[Count[primeMS[#],?EvenQ]]&&OddQ[Count[primeMS[#],?OddQ]]&]
Previous Showing 21-26 of 26 results.

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