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 31-35 of 35 results.

A341447 Heinz numbers of integer partitions whose only even part is the smallest.

Original entry on oeis.org

3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317
Offset: 1

Views

Author

Gus Wiseman, Feb 13 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only even prime index (counting multiplicity) is the smallest.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      3: (2)         77: (5,4)     165: (5,3,2)
      7: (4)         79: (22)      173: (40)
     13: (6)         89: (24)      177: (17,2)
     15: (3,2)       93: (11,2)    181: (42)
     19: (8)        101: (26)      193: (44)
     29: (10)       107: (28)      199: (46)
     33: (5,2)      113: (30)      201: (19,2)
     37: (12)       119: (7,4)     217: (11,4)
     43: (14)       123: (13,2)    219: (21,2)
     51: (7,2)      131: (32)      221: (7,6)
     53: (16)       139: (34)      223: (48)
     61: (18)       141: (15,2)    229: (50)
     69: (9,2)      151: (36)      239: (52)
     71: (20)       161: (9,4)     249: (23,2)
     75: (3,3,2)    163: (38)      251: (54)

Crossrefs

These partitions are counted by A087897, shifted left once.
Terms of A340933 can be factored into elements of this sequence.
The odd version is A341446.
A000009 counts partitions into odd parts, ranked by A066208.
A001222 counts prime factors.
A005843 lists even numbers.
A026804 counts partitions whose least part is odd, ranked by A340932.
A026805 counts partitions whose least part is even, ranked by A340933.
A027187 counts partitions with even length/max, ranked by A028260/A244990.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058696 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A066207 lists numbers with all even prime indices.
A112798 lists the prime indices of each positive integer.
Cf. A026804, A035363, A036554, A160786, A244991, A257991, A257992, A300272, A300063, A340854/A340855.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]

A103421 Number of compositions of n in which the greatest part is odd.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 30, 62, 129, 263, 534, 1076, 2160, 4318, 8612, 17145, 34097, 67764, 134638, 267506, 531606, 1056812, 2101854, 4182462, 8327263, 16588973, 33066080, 65945522, 131588128, 262702054, 524699094, 1048433468, 2095744336
Offset: 1

Views

Author

Vladeta Jovovic, Feb 04 2005

Keywords

Crossrefs

Cf. A103419, A103420, A103422, A027187, A027193, A026804, A026805.

Programs

  • Mathematica
    Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n - 1)/((1 - 2x + x^(2n - 1))*(1 - 2x + x^(2n))), {n, 35}]], {x, 0, 35}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

Formula

G.f.: Sum((1-x)^2*x^(2*n-1)/((1-2*x+x^(2*n-1))*(1-2*x+x^(2*n))), n=1..infinity).
a(n) + A103422(n) = 2^(n-1). - R. J. Mathar, Mar 24 2018

Extensions

More terms from Robert G. Wilson v, Feb 05 2005

A366532 Heinz numbers of integer partitions with at least one even and odd part.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 30, 33, 35, 36, 38, 42, 45, 48, 51, 52, 54, 56, 58, 60, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 90, 93, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 114, 116, 119, 120, 122, 123, 126, 130, 132, 135, 138, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

These partitions are counted by A006477.
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 terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Crossrefs

These partitions are counted by A006477.
Just even: A324929, counted by A047967.
Just odd: A366322, counted by A086543 (even bisection of A182616).
A031368 lists primes of odd index, even A031215.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 lists prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
A257992 counts even prime indices, distinct A324967.
Cf. A000720, A001222, A003963, A026804, A244991, A257992.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or@@EvenQ/@prix[#]&&Or@@OddQ/@prix[#]&]

Formula

Intersection of A324929 and A366322.

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 31-35 of 35 results.

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

Content is available under The OEIS End-User License Agreement.