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 11-17 of 17 results.

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.

A340931 Heinz numbers of integer partitions of odd numbers into an odd number of parts.

Original entry on oeis.org

2, 5, 8, 11, 17, 18, 20, 23, 31, 32, 41, 42, 44, 45, 47, 50, 59, 67, 68, 72, 73, 78, 80, 83, 92, 97, 98, 99, 103, 105, 109, 110, 114, 124, 125, 127, 128, 137, 149, 153, 157, 162, 164, 167, 168, 170, 174, 176, 179, 180, 182, 188, 191, 195, 197, 200, 207, 211
Offset: 1

Views

Author

Gus Wiseman, Feb 05 2021

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with the corresponding partitions begins:
      2: (1)             50: (3,3,1)        109: (29)
      5: (3)             59: (17)           110: (5,3,1)
      8: (1,1,1)         67: (19)           114: (8,2,1)
     11: (5)             68: (7,1,1)        124: (11,1,1)
     17: (7)             72: (2,2,1,1,1)    125: (3,3,3)
     18: (2,2,1)         73: (21)           127: (31)
     20: (3,1,1)         78: (6,2,1)        128: (1,1,1,1,1,1,1)
     23: (9)             80: (3,1,1,1,1)    137: (33)
     31: (11)            83: (23)           149: (35)
     32: (1,1,1,1,1)     92: (9,1,1)        153: (7,2,2)
     41: (13)            97: (25)           157: (37)
     42: (4,2,1)         98: (4,4,1)        162: (2,2,2,2,1)
     44: (5,1,1)         99: (5,2,2)        164: (13,1,1)
     45: (3,2,2)        103: (27)           167: (39)
     47: (15)           105: (4,3,2)        168: (4,2,1,1,1)

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A160786.
The even version is A236913 (A340784).
The case of where the prime indices are also odd is A300272.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts odd-length partitions (A026424).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
Cf. A000041, A316413, A326845, A340385, A340386, A340387, A340604, A340607, A340854, A340855.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[PrimeOmega[#]]&&OddQ[Total[primeMS[#]]]&]

Formula

Intersection of A026424 and A300063.

A340689 Numbers with a factorization of length 2^k into factors > 1, where k is the greatest factor.

Original entry on oeis.org

1, 16, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 131072, 196608, 262144, 294912, 393216, 442368, 524288, 589824, 663552, 786432, 884736, 995328, 1048576, 1179648, 1327104, 1492992, 1572864, 1769472, 1990656, 2097152, 2239488, 2359296, 2654208, 2985984, 3145728
Offset: 1

Views

Author

Gus Wiseman, Jan 28 2021

Keywords

Examples

			The initial terms and a valid factorization of each are:
         1 =
        16 = 2*2*2*2
       384 = 2*2*2*2*2*2*2*3
       576 = 2*2*2*2*2*2*3*3
       864 = 2*2*2*2*2*3*3*3
      1296 = 2*2*2*2*3*3*3*3
      1944 = 2*2*2*3*3*3*3*3
      2916 = 2*2*3*3*3*3*3*3
      4374 = 2*3*3*3*3*3*3*3
      6561 = 3*3*3*3*3*3*3*3
    131072 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*4
    196608 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*3*4
    262144 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*4*4
    294912 = 2*2*2*2*2*2*2*2*2*2*2*2*2*3*3*4

Links

Crossrefs

Partitions of the prescribed type are counted by A340611.
The conjugate version is A340690.
A001055 counts factorizations, with strict case A045778.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],Length[#]==2^Max@@#&]!={}&]

Extensions

More terms from Chai Wah Wu, Feb 01 2021

A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

Original entry on oeis.org

2, 8, 12, 16, 32, 48, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 224, 240, 256, 280, 288, 320, 336, 384, 392, 432, 448, 480, 512, 576, 640, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1008, 1024, 1056, 1120, 1152, 1200, 1248, 1280, 1296, 1344
Offset: 1

Views

Author

Gus Wiseman, Jan 28 2021

Keywords

Examples

			The initial terms and a valid factorization of each:
      2 = 2           168 = 3*7*8        512 = 2*2*2*2*32
      8 = 2*4         192 = 2*2*3*16     576 = 2*2*9*16
     12 = 3*4         200 = 5*5*8        640 = 2*2*10*16
     16 = 4*4         224 = 4*7*8        672 = 2*3*7*16
     32 = 2*2*8       240 = 5*6*8        704 = 2*2*11*16
     48 = 2*3*8       256 = 2*2*4*16     720 = 3*3*5*16
     64 = 2*4*8       280 = 5*7*8        768 = 2*2*2*3*32
     72 = 3*3*8       288 = 2*3*3*16     800 = 2*5*5*16
     80 = 2*5*8       320 = 2*2*5*16     832 = 2*2*13*16
     96 = 2*6*8       336 = 6*7*8        864 = 2*3*9*16
    112 = 2*7*8       384 = 2*2*6*16     896 = 2*2*14*16
    120 = 3*5*8       392 = 7*7*8        960 = 2*2*15*16
    128 = 2*2*2*16    432 = 3*3*3*16    1008 = 3*3*7*16
    144 = 3*6*8       448 = 2*2*7*16    1024 = 2*2*2*4*32
    160 = 4*5*8       480 = 2*3*5*16    1056 = 2*3*11*16

Crossrefs

Partitions of the prescribed type are counted by A340611.
The conjugate version is A340689.
A001055 counts factorizations, with strict case A045778.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],2^Length[#]==Max@@#&]!={}&]

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937
Offset: 0

Views

Author

Gus Wiseman, Aug 01 2021

Keywords

Examples

			The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

Links

Crossrefs

Odd bisection of A067659, which is ranked by A030059.
The even version is the even bisection of A067661.
The case of all odd parts is counted by A069911 (non-strict: A078408).
The non-strict version is A160786, ranked by A340931.
The non-strict even version is A236913, ranked by A340784.
The even-length version is A343942 (non-strict: A236914).
The even-sum version is A344650 (non-strict: A236559 or A344611).
A000009 counts partitions with all odd parts, ranked by A066208.
A000009 counts strict partitions, ranked by A005117.
A027193 counts odd-length partitions, ranked by A026424.
A027193 counts odd-maximum partitions, ranked by A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Other cases of odd length:
- A024429 set partitions
- A089677 ordered set partitions
- A166444 compositions
- A174726 ordered factorizations
- A332304 strict compositions
- A339890 factorizations
Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021
  • Mathematica
    Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]

Extensions

More terms from Alois P. Heinz, Aug 05 2021

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]]&]

A341449 Heinz numbers of integer partitions into odd parts > 1.

Original entry on oeis.org

1, 5, 11, 17, 23, 25, 31, 41, 47, 55, 59, 67, 73, 83, 85, 97, 103, 109, 115, 121, 125, 127, 137, 149, 155, 157, 167, 179, 187, 191, 197, 205, 211, 227, 233, 235, 241, 253, 257, 269, 275, 277, 283, 289, 295, 307, 313, 331, 335, 341, 347, 353, 365, 367, 379, 389
Offset: 1

Views

Author

Gus Wiseman, Feb 15 2021

Keywords

Comments

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 together with their Heinz numbers begins:
      1: ()        97: (25)       197: (45)       307: (63)
      5: (3)      103: (27)       205: (13,3)     313: (65)
     11: (5)      109: (29)       211: (47)       331: (67)
     17: (7)      115: (9,3)      227: (49)       335: (19,3)
     23: (9)      121: (5,5)      233: (51)       341: (11,5)
     25: (3,3)    125: (3,3,3)    235: (15,3)     347: (69)
     31: (11)     127: (31)       241: (53)       353: (71)
     41: (13)     137: (33)       253: (9,5)      365: (21,3)
     47: (15)     149: (35)       257: (55)       367: (73)
     55: (5,3)    155: (11,3)     269: (57)       379: (75)
     59: (17)     157: (37)       275: (5,3,3)    389: (77)
     67: (19)     167: (39)       277: (59)       391: (9,7)
     73: (21)     179: (41)       283: (61)       401: (79)
     83: (23)     187: (7,5)      289: (7,7)      415: (23,3)
     85: (7,3)    191: (43)       295: (17,3)     419: (81)

Crossrefs

Note: A-numbers of ranking sequences are in parentheses below.
Partitions with no ones are A002865 (A005408).
The case of even parts is A035363 (A066207).
These partitions are counted by A087897.
The version for factorizations is A340101.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A056239 adds up prime indices.
A078408 counts partitions with odd parts, length, and sum (A300272).
A112798 lists the prime indices of each positive integer.
A257991/A257992 count odd/even prime indices.
Cf. A026804 (A340932), A160786 (A340931), A340386, A340604, A340607, A340785, A340854/A340855, A341446.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&OddQ[Times@@primeMS[#]]&]
Previous Showing 11-17 of 17 results.

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