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.

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A340788 Heinz numbers of integer partitions of negative rank.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405
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)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 is (A340929).
The even case is A101708 is (A340930).
The positive version is (A340787).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson 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.
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), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]

Formula

For all terms A061395(a(n)) < A001222(a(n)).

A340605 Heinz numbers of integer partitions of even positive rank.

Original entry on oeis.org

5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2021

Keywords

Comments

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of partitions with their Heinz numbers begins:
      5: (3)         57: (8,2)       97: (25)
     11: (5)         58: (10,1)      99: (5,2,2)
     14: (4,1)       59: (17)       102: (7,2,1)
     17: (7)         65: (6,3)      103: (27)
     21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
     23: (9)         67: (19)       106: (16,1)
     26: (6,1)       68: (7,1,1)    109: (29)
     31: (11)        73: (21)       110: (5,3,1)
     35: (4,3)       74: (12,1)     111: (12,2)
     38: (8,1)       83: (23)       122: (18,1)
     39: (6,2)       86: (14,1)     124: (11,1,1)
     41: (13)        87: (10,2)     127: (31)
     44: (5,1,1)     91: (6,4)      129: (14,2)
     47: (15)        92: (9,1,1)    133: (8,4)
     49: (4,4)       95: (8,3)      137: (33)

Links

Crossrefs

Note: Heinz numbers are given in parentheses below.
Allowing any positive rank gives A064173 (A340787).
The odd version is counted by A101707 (A340604).
These partitions are counted by A101708.
The not necessarily positive case is counted by A340601 (A340602).
A001222 counts prime indices.
A061395 gives maximum prime index.
A072233 counts partitions by sum and length.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340692 counts partitions of odd rank (A340603).
- Even -
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.

Programs

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

Formula

A061395(a(n)) - A001222(a(n)) is even and positive.
A340604 \/ A340605 = A340787.

A340603 Heinz numbers of integer partitions of odd rank.

Original entry on oeis.org

3, 4, 7, 10, 12, 13, 15, 16, 18, 19, 22, 25, 27, 28, 29, 33, 34, 37, 40, 42, 43, 46, 48, 51, 52, 53, 55, 60, 61, 62, 63, 64, 69, 70, 71, 72, 76, 77, 78, 79, 82, 85, 88, 89, 90, 93, 94, 98, 100, 101, 105, 107, 108, 112, 113, 114, 115, 116, 117, 118, 119, 121
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2021

Keywords

Comments

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of partitions with their Heinz numbers begins:
      3: (2)           33: (5,2)           63: (4,2,2)
      4: (1,1)         34: (7,1)           64: (1,1,1,1,1,1)
      7: (4)           37: (12)            69: (9,2)
     10: (3,1)         40: (3,1,1,1)       70: (4,3,1)
     12: (2,1,1)       42: (4,2,1)         71: (20)
     13: (6)           43: (14)            72: (2,2,1,1,1)
     15: (3,2)         46: (9,1)           76: (8,1,1)
     16: (1,1,1,1)     48: (2,1,1,1,1)     77: (5,4)
     18: (2,2,1)       51: (7,2)           78: (6,2,1)
     19: (8)           52: (6,1,1)         79: (22)
     22: (5,1)         53: (16)            82: (13,1)
     25: (3,3)         55: (5,3)           85: (7,3)
     27: (2,2,2)       60: (3,2,1,1)       88: (5,1,1,1)
     28: (4,1,1)       61: (18)            89: (24)
     29: (10)          62: (11,1)          90: (3,2,2,1)

Links

Crossrefs

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A340692.
The complement is A340602, counted by A340601.
The case of positive rank is A340604.
- Rank -
A001222 gives number of prime indices.
A047993 counts partitions of rank 0 (A106529).
A061395 gives maximum prime index.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340608, A340609, A340610.

Programs

  • Mathematica
    Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]

Formula

A061395(a(n)) - A001222(a(n)) is odd.

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95
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:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson 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), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>PrimeOmega[#]&]

Formula

For all terms A061395(a(n)) > A001222(a(n)).

A101200 Number of partitions of n with rank 3 (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 3, 6, 7, 10, 11, 17, 18, 26, 30, 40, 47, 63, 72, 94, 111, 140, 165, 209, 244, 304, 359, 440, 519, 634, 743, 901, 1060, 1273, 1494, 1789, 2092, 2491, 2914, 3449, 4026, 4752, 5530, 6502, 7561, 8852, 10272, 11997, 13889, 16171, 18695, 21700, 25041, 29002
Offset: 1

Views

Author

Emeric Deutsch, Dec 12 2004

Keywords

Comments

Column k=3 in the triangle A063995.

Examples

			a(6)=1 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.

References

  • George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Links

Crossrefs

Cf. A000041, A063995, A101198, A101199.

Programs

  • Maple
    with(combinat): for n from 1 to 45 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=3 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);
  • Mathematica
    Table[Count[IntegerPartitions[n],?(#[[1]]-Length[#]==3&)],{n,60}] (* _Harvey P. Dale, Feb 11 2025 *)

Extensions

More terms, Joerg Arndt, Oct 07 2012

A123975 Number of Garden of Eden partitions of n in Bulgarian Solitaire.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 147, 190, 243, 311, 394, 499, 627, 786, 980, 1220, 1510, 1865, 2294, 2816, 3443, 4202, 5110, 6203, 7507, 9067, 10923, 13135, 15755, 18865, 22540, 26885, 32001, 38032, 45112, 53430, 63171
Offset: 1

Views

Author

Vladeta Jovovic, Nov 23 2006

Keywords

Comments

a(n) gives the number of times n occurs in A225794. - Antti Karttunen, Jul 27 2013

Links

Crossrefs

Cf. A000041, A045943, A064173, A101198, A237831.
Cf. A227753, A225794, A226062.

Programs

  • Maple
    p:=product(1/(1-q^i), i=1..200)*sum((-1)^(r-1)*q^((3*r^2+3*r)/2), r=1..200):s:=series(p, q, 200): for j from 0 to 199 do printf(`%d,`,coeff(s, q, j)) od: # James Sellers, Nov 30 2006
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k+1)/2)))) \\ Seiichi Manyama, May 21 2023

Formula

a(n) = A064173(n) - A101198(n).
a(n) = Sum_{j>=1} (-1)^(j+1)*p(n-b(j)) where b(j) = 3*j*(j+1)/2 (A045943) and p(n) is the number of partitions of n (see A000041). See Hopkins & Sellers. - Michel Marcus, Sep 26 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 19*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

Extensions

More terms from James Sellers, Nov 30 2006

A101709 Number of partitions of n having nonnegative even rank (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

1, 0, 2, 1, 3, 2, 7, 5, 11, 10, 20, 20, 34, 35, 57, 62, 92, 104, 151, 171, 237, 274, 371, 433, 571, 670, 870, 1025, 1306, 1543, 1947, 2299, 2864, 3387, 4183, 4943, 6052, 7143, 8688, 10242, 12371, 14566, 17503, 20567, 24583, 28841, 34319, 40188, 47618, 55654, 65700, 76643, 90149, 104968
Offset: 1

Views

Author

Emeric Deutsch, Dec 12 2004

Keywords

Comments

A101709(n)=A101707(n)+A047993(n). A000041(n)=2*A101707(n)+A101708(n)+A101709(n).

Examples

			a(5)=3 because the partitions of 5 with nonnegative even ranks are 5 (rank=4), 41 (rank=2) and 311 (rank=0).

References

  • George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Crossrefs

Cf. A000041, A101707, A101708, A047993.
Cf. A101198-A101200.

Formula

G.f.: Sum((-1)^(k+1)*x^((3*k^2-k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004

Extensions

More terms, Joerg Arndt, Oct 07 2012

A101199 Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 2, 3, 3, 6, 6, 9, 10, 15, 16, 23, 27, 36, 42, 55, 64, 84, 98, 124, 147, 185, 217, 270, 318, 391, 461, 562, 661, 802, 942, 1132, 1331, 1592, 1864, 2220, 2597, 3077, 3593, 4240, 4940, 5811, 6758, 7916, 9192, 10737, 12438, 14488, 16755, 19459, 22465, 26024, 29987
Offset: 1

Views

Author

Emeric Deutsch, Dec 12 2004

Keywords

Comments

Column k=2 in the triangle A063995.

Examples

			a(6)=1 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.

References

  • George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Links

Crossrefs

Cf. A000041, A063995, A101198, A101200.

Programs

  • Maple
    with(combinat): for n from 1 to 45 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=2 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);
  • Mathematica
    Table[Count[Max[#]-Length[#]&/@IntegerPartitions[n],2],{n,60}] (* Harvey P. Dale, Dec 22 2018 *)

Formula

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 26 2023

Extensions

More terms from Joerg Arndt, Oct 07 2012

A325234 Heinz numbers of integer partitions with Dyson rank -1.

Original entry on oeis.org

4, 12, 18, 27, 40, 60, 90, 100, 112, 135, 150, 168, 225, 250, 252, 280, 352, 375, 378, 392, 420, 528, 567, 588, 625, 630, 700, 792, 832, 880, 882, 945, 980, 1050, 1188, 1232, 1248, 1320, 1323, 1372, 1470, 1575, 1750, 1782, 1848, 1872, 1936, 1980, 2058, 2080
Offset: 1

Views

Author

Gus Wiseman, Apr 13 2019

Keywords

Comments

Numbers whose maximum prime index is one fewer than their number of prime indices counted with multiplicity.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
     4: {1,1}
    12: {1,1,2}
    18: {1,2,2}
    27: {2,2,2}
    40: {1,1,1,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   100: {1,1,3,3}
   112: {1,1,1,1,4}
   135: {2,2,2,3}
   150: {1,2,3,3}
   168: {1,1,1,2,4}
   225: {2,2,3,3}
   250: {1,3,3,3}
   252: {1,1,2,2,4}
   280: {1,1,1,3,4}
   352: {1,1,1,1,1,5}
   375: {2,3,3,3}
   378: {1,2,2,2,4}
   392: {1,1,1,4,4}

Links

Crossrefs

Positions of -1's in A257541.
Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325235.

Programs

  • Mathematica
    Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==-1&]

A325235 Heinz numbers of integer partitions with Dyson rank 1 or -1.

Original entry on oeis.org

3, 4, 10, 12, 15, 18, 25, 27, 28, 40, 42, 60, 63, 70, 88, 90, 98, 100, 105, 112, 132, 135, 147, 150, 168, 175, 198, 208, 220, 225, 245, 250, 252, 280, 297, 308, 312, 330, 343, 352, 375, 378, 392, 420, 462, 468, 484, 495, 520, 528, 544, 550, 567, 588, 625, 630
Offset: 1

Views

Author

Gus Wiseman, Apr 13 2019

Keywords

Comments

Numbers whose maximum prime index and number of prime indices counted with multiplicity differ by 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  105: {2,3,4}
  112: {1,1,1,1,4}

Links

Crossrefs

Positions of 1's and -1's in A257541.
Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325234.

Programs

  • Mathematica
    Select[Range[1000],Abs[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]==1&]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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