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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A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394
Offset: 0

Views

Author

Gus Wiseman, Jan 31 2022

Keywords

Comments

These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

Examples

			The a(1) = 1 through a(9) = 7 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    221    33      331      44        333
              1111  11111  222     22111    332       441
                           2211    1111111  2222      22221
                           111111           3311      33111
                                            221111    2211111
                                            11111111  111111111

Crossrefs

The ordered version (compositions) is A016116.
The even-length case is A035363.
A reverse version is A096441, both A349060.
The version for unequal instead of equal is A122129, even-length A351008.
The version for even instead of odd indices is A351003, even-length A351012.
The strict version is A351005, opposite A351006, even-length A035457.
Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497
Offset: 0

Views

Author

Gus Wiseman, Feb 12 2022

Keywords

Examples

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Links

Crossrefs

The version for run-lengths instead of runs is A144300.
The version for normal multisets is A283353.
The Heinz numbers of these partitions are A351201.
The complement is counted by A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions, ranked by A333489.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]
  • Python
    from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n,size=True):
        for q in permutations(Counter(p).elements(),s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

Formula

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

Extensions

a(26) onwards from Andrew Howroyd, Jan 27 2024

A373405 Sum of the n-th maximal antirun of odd primes differing by more than two.

Original entry on oeis.org

3, 5, 18, 30, 71, 109, 202, 199, 522, 210, 617, 288, 990, 372, 390, 860, 701, 1281, 829, 1194, 1645, 4578, 852, 2682, 4419, 3300, 2927, 2438, 1891, 2602, 14660, 1632, 1650, 3378, 3480, 18141, 2052, 3121, 2112, 4310, 8922, 13131, 6253, 3851, 3889, 3929, 13788
Offset: 1

Views

Author

Gus Wiseman, Jun 05 2024

Keywords

Comments

The length of this run is given by A027833 (except initial term).
An antirun of a sequence (in this case A000040\{2}) is an interval of positions at which consecutive terms differ by more than one.

Examples

			Row-sums of:
   3
   5
   7  11
  13  17
  19  23  29
  31  37  41
  43  47  53  59
  61  67  71
  73  79  83  89  97 101

Links

Crossrefs

The partial sums are a subset of A071148 (partial sums of odd primes).
Functional neighbors: A001359, A006512, A027833 (partial sums A029707), A373404, A373406, A373411, A373412.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
Cf. A025157, A038664, A046933, A293697, A350842 (strict A350844), A371201.

Programs

  • Mathematica
    Total/@Split[Select[Range[3,1000],PrimeQ],#1+2!=#2&]//Most

A373406 Sum of the n-th maximal run of odd primes differing by two.

Original entry on oeis.org

15, 24, 36, 23, 60, 37, 84, 47, 53, 120, 67, 144, 79, 83, 89, 97, 204, 216, 113, 127, 131, 276, 300, 157, 163, 167, 173, 360, 384, 396, 211, 223, 456, 233, 480, 251, 257, 263, 540, 277, 564, 293, 307, 624, 317, 331, 337, 696, 353, 359, 367, 373, 379, 383
Offset: 1

Views

Author

Gus Wiseman, Jun 05 2024

Keywords

Comments

The length of this run is given by A251092.
For this sequence we define a run to be an interval of positions at which consecutive terms differ by two. Normally, a run has consecutive terms differing by one, but odd prime numbers already differ by at least two.
Contains A054735 (sums of twin prime pairs) without its first two terms and A007510 (non-twin primes) as subsequences. - R. J. Mathar, Jun 07 2024

Examples

			Row-sums of:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
  83
  89
  97

Links

Crossrefs

The partial sums are a subset of A071148 (partial sums of odd primes).
Functional neighbors: A025584, A054265, A067774, A251092 (or A175632), A373405, A373413, A373414.
A000040 lists the primes, differences A001223.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005117, A007053, A029707, A038664, A293697, A350842, A371201.

Programs

  • Mathematica
    Total/@Split[Select[Range[3,100],PrimeQ],#1+2==#2&]//Most

A373412 Sum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

12, 99, 52, 180, 93, 49, 335, 279, 156, 629, 99, 540, 237, 245, 125, 521, 567, 450, 963, 340, 347, 728, 1386, 1080, 1637, 243, 244, 1511, 1610, 555, 852, 1171, 2142, 960, 985, 1689, 343, 1042, 351, 1068, 724, 732, 1116, 1905, 1980, 2898, 424, 2161, 3150, 2339
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2024

Keywords

Comments

The length of this antirun is given by A373409.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

Examples

			Row-sums of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

Links

Crossrefs

The partial sums are a subset of A329472.
Functional neighbors: A068781, A373404, A373405, A373409, A373410, A373411, A373414.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A049094, A061399, A101836, A112925, A143658, A294242, A350842, A372683, A373197.

Programs

  • Mathematica
    Total/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]//Most

A373410 Minimum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

4, 9, 25, 28, 45, 49, 50, 64, 76, 81, 99, 100, 117, 121, 125, 126, 136, 148, 153, 169, 172, 176, 189, 208, 225, 243, 244, 245, 261, 276, 280, 289, 297, 316, 325, 333, 343, 344, 351, 352, 361, 364, 369, 376, 388, 405, 424, 425, 441, 460, 476, 477, 496, 508, 513
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2024

Keywords

Comments

The maximum is given by A068781.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Consists of 4 and all nonsquarefree numbers n such that n - 1 is also nonsquarefree.

Examples

			Row-minima of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

Links

Crossrefs

Functional neighbors: A005381, A006512, A053806, A068781, A373408, A373409, A373412.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A049094, A061399, A077641, A077643, A112926, A143658, A294242, A350842, A372683, A372684, A373125.

Programs

  • Mathematica
    First/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]

Formula

a(1) = 4; a(n>1) = A068781(n-1) + 1.

A350841 Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 63, 68, 76, 80, 84, 88, 92, 99, 100, 104, 112, 116, 117, 124, 126, 132, 136, 140, 148, 152, 153, 156, 160, 164, 168, 171, 172, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 244, 248, 252, 260, 261
Offset: 1

Views

Author

Gus Wiseman, Jan 26 2022

Keywords

Comments

We define a difference of a partition to be a difference of two adjacent parts.

Examples

			The terms together with their prime indices begin:
   20: (3,1,1)
   28: (4,1,1)
   40: (3,1,1,1)
   44: (5,1,1)
   52: (6,1,1)
   56: (4,1,1,1)
   63: (4,2,2)
   68: (7,1,1)
   76: (8,1,1)
   80: (3,1,1,1,1)
   84: (4,2,1,1)
   88: (5,1,1,1)
   92: (9,1,1)
   99: (5,2,2)

Crossrefs

Heinz number rankings are in parentheses below.
Taking just one condition gives (A073492) and (A065201), counted by A239955.
These partitions are counted by A350839.
A000041 = integer partitions, strict A000009.
A034296 = partitions with no gaps (A073491), strict A001227 (A073485).
A090858 = partitions with a single gap of size 1 (A325284).
A116931 = partitions with no successions (A319630), strict A003114.
A116932 = partitions with no successions or gaps of size 1, strict A025157.
A350842 = partitions with no gaps of size 1, strict A350844, sets A005314.
Cf. A000070, A024619, A026424, A055932, A183558, A277103, A305148, A321440, A350838.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],(Min@@Differences[Reverse[primeMS[#]]]<-1)&&(Min@@Differences[conj[primeMS[#]]]<-1)&]

A351008 Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093
Offset: 0

Views

Author

Gus Wiseman, Jan 31 2022

Keywords

Comments

Write the series in the g.f. given below as Sum_{k >= 0} q^(1 + 3 + 5 + ... + 2*k-1 + 2*k)/Product_{i = 1..2*k} 1 - q^i. Since 1/Product_{i = 1..2*k} 1 - q^i is the g.f. for partitions with parts <= 2*k, we see that the k-th summand of the series is the g.f. for partitions with largest part 2*k in which every odd number less than 2*k appears at least once as a part. The partitions of this type are conjugate to (and hence equinumerous with) the partitions (y_1, y_2, ..., y_{2*k}) of even length 2*k having strict decrease y_i > y_(i+1) for all odd i < 2*k. - Peter Bala, Jan 02 2024

Examples

			The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):
  21   31   32   42   43   53     54     64     65     75     76
            41   51   52   62     63     73     74     84     85
                      61   71     72     82     83     93     94
                           3221   81     91     92     A2     A3
                                  4221   4321   A1     B1     B2
                                         5221   4331   4332   C1
                                                5321   5331   5332
                                                6221   5421   5431
                                                       6321   6331
                                                       7221   6421
                                                              7321
                                                              8221
a(10) = 6: the six partitions 64, 73, 82, 91, 4321 and 5221 listed above have conjugate partitions 222211, 2221111, 22111111, 211111111, 4321 and 43111, These are the partitions of 10 with largest part L even and such that every odd number less than L appears at least once as a part. - _Peter Bala_, Jan 02 2024

Crossrefs

The version for equal instead of unequal is A035363.
The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A122129, opposite A122135.
The odd-length version appears to be A122130.
The alternately unequal and equal version is A351007, any length A351006.
Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.

Programs

  • Maple
    series(add(q^(n*(n+2))/mul(1 - q^k, k = 1..2*n), n = 0..10), q, 141):
seq(coeftayl(%, q = 0, n), n = 0..140); # Peter Bala, Jan 03 2025
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

Formula

Conjecture: a(n+1) = A122129(n+1) - A122130(n). - Gus Wiseman, Feb 21 2022
G.f.: Sum_{n >= 0} q^(n*(n+2))/Product_{k = 1..2*n} 1 - q^k = 1 + q^3 + q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 4*q^8 + 5*q^9 + 6*q^10 + .... - Peter Bala, Jan 02 2024

A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 18, 24, 31, 41, 53, 70, 87, 112, 140, 178, 221, 277, 344, 428, 526, 648, 792, 971, 1180, 1436, 1738, 2103, 2533, 3049, 3660, 4387, 5242, 6259, 7450, 8860, 10511, 12453, 14723, 17387, 20489, 24121, 28343, 33269, 38982, 45632, 53327
Offset: 0

Views

Author

Gus Wiseman, Jan 18 2022

Keywords

Comments

The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).

Examples

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (32)     (33)      (43)
                    (31)    (41)     (51)      (52)
                    (1111)  (311)    (222)     (61)
                            (11111)  (411)     (322)
                                     (3111)    (331)
                                     (111111)  (511)
                                               (4111)
                                               (31111)
                                               (1111111)

Links

Crossrefs

The version with quotients >= 2 is A000929, sets A018819.
<= 2 is A342094, ranked by A342191.
< 2 is A342096, sets A045690, strict A342097.
> 2 is A342098, sets A040039.
The sets version (subsets of prescribed maximum) is A045691.
These partitions are ranked by A350838.
The strict case is A350840.
A version for differences is A350842, strict A350844.
The complement is counted by A350846, ranked by A350845.
A000041 = integer partitions.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free partitions, ranked by A320340.
Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83
Offset: 1

Views

Author

Gus Wiseman, Jan 18 2022

Keywords

Comments

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

Examples

			The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

Crossrefs

The version with quotients >= 2 is counted by A000929, sets A018819.
<= 2 is A342191, counted by A342094.
< 2 is counted by A342096, sets A045690.
> 2 is counted by A342098, sets A040039.
The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
For differences instead of quotients we have A350842, strict A350844.
The complement is A350845, counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]
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