A115029 -id:A115029 - cp's OEIS frontend

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

1, 1, 2, 4, 7, 13, 25, 46, 85, 160, 301, 561, 1056, 1984, 3730, 7037, 13273, 25056, 47382, 89666, 169833, 322038, 611128, 1160660, 2206219, 4196730, 7988731, 15217557, 29005987, 55321015, 105570219, 201569648, 385059094, 735929616, 1407145439, 2691681402

Offset: 0

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

John Tyler Rascoe, Table of n, a(n) for n = 0..100
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296 or A115029.
These compositions have ranks A374519.
The complement is counted by A374640.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly decreasing runs we have A374742, ranks A374741.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Other types of run-leaders (instead of identical):
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A189076, A238343, A333213, A373949, A374515, A374518.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]

C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
D_x(30) \\ John Tyler Rascoe, Aug 16 2024

G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - John Tyler Rascoe, Aug 16 2024

a(26) onwards from John Tyler Rascoe, Aug 16 2024

A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 17, 23, 29, 38, 47, 60, 74, 93, 113, 141, 171, 211, 253, 309, 370, 447, 532, 639, 758, 904, 1066, 1265, 1487, 1754, 2053, 2411, 2813, 3289, 3823, 4454, 5161, 5990, 6920, 8005, 9223, 10634, 12218, 14048, 16101, 18462, 21107

Offset: 0

Gus Wiseman, Aug 14 2024

nonn

An anti-run is a sequence with no adjacent equal parts.
These are partitions with no part appearing more than twice and greatest part appearing only once.
Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.

			The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
                (21)  (31)   (32)   (42)   (43)    (53)    (54)
                      (211)  (41)   (51)   (52)    (62)    (63)
                             (311)  (321)  (61)    (71)    (72)
                                    (411)  (322)   (422)   (81)
                                           (421)   (431)   (432)
                                           (511)   (521)   (522)
                                           (3211)  (611)   (531)
                                                   (3221)  (621)
                                                   (4211)  (711)
                                                           (4221)
                                                           (4311)
                                                           (5211)
                                                           (32211)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Includes all strict partitions A000009.
For identical instead of distinct see: A034296, A115029, A374760, A374759.
For compositions instead of partitions we have A374761.
For minima instead of maxima we have A375134, ranks A375398.
The complement is counted by A375401, ranks A375403.
These partitions are ranked by A375402, for compositions A374767.
The complement for minima instead of maxima is A375404, ranks A375399.
A000041 counts integer partitions.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A141199, A279790, A358830, A358833, A358836, A358905, A374704, A374757, A374758, A375136, A375400.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]

A_x(N) = {my(x='x+O('x^N), f=sum(i=0,N,(x^i)*prod(j=1,i-1,(1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024

G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024

A375134 Number of integer partitions of n whose maximal anti-runs have distinct minima.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 18, 21, 28, 33, 43, 52, 66, 78, 98, 116, 145, 171, 209, 247, 300, 352, 424, 499, 595, 695, 826, 963, 1138, 1322, 1553, 1802, 2106, 2435, 2835, 3271, 3795, 4365, 5046, 5792, 6673, 7641, 8778, 10030, 11490, 13099, 14968, 17030

Offset: 0

Gus Wiseman, Aug 14 2024

nonn

These are partitions with no part appearing more than twice and with the least part appearing only once.

Also the number of reversed integer partitions of n whose maximal anti-runs have distinct minima.

			The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with minima (5,3,1), so y is counted under a(29).
The a(1) = 1 through a(9) = 11 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (12)  (13)  (14)   (15)   (16)   (17)    (18)
                        (23)   (24)   (25)   (26)    (27)
                        (122)  (123)  (34)   (35)    (36)
                                      (124)  (125)   (45)
                                      (133)  (134)   (126)
                                             (233)   (135)
                                             (1223)  (144)
                                                     (234)
                                                     (1224)
                                                     (1233)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Includes all strict partitions A000009.
For identical instead of distinct leaders we have A115029.
A version for compositions instead of partitions is A374518, ranks A374638.
For minima instead of maxima we have A375133, ranks A375402.
These partitions have ranks A375398.
The complement is counted by A375404, ranks A375399.
A000041 counts integer partitions.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A034296, A141199, A358830, A358836, A358905, A374704, A374757, A374758, A374761, A375136, A375400, A375401.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]

A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,(x^i)*prod(j=i+1,N-i,(1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024

G.f.: 1 + Sum_{i>0} (x^i * Product_{j>i} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024

A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 59, 114, 222, 434, 844, 1641, 3189, 6192, 12020, 23320, 45213, 87624, 169744, 328684, 636221, 1231067, 2381269, 4604713, 8901664

Offset: 0

Gus Wiseman, Aug 01 2024

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For reversed partitions instead of compositions we have A115029.
The complement is A374699.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A189076, complement A374636.
- For leaders of weakly decreasing runs we have A374747.
- For leaders of strictly decreasing runs we have A374765.
- For leaders of strictly increasing runs we have A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]

A375398 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are distinct.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

First differs from A375402 in lacking 20.
An anti-run is a sequence with no adjacent equal parts.
The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
Note the prime factors can alternatively be taken in weakly decreasing order.

			The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs ((2),(2,3,5),(5)), with minima (2,2,5), so 300 is not in the sequence.
The prime factors of 450 are {2,3,3,5,5}, with maximal anti-runs ((2,3),(3,5),(5)), with minima (2,3,5), so 450 is in the sequence.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

A version for compositions is A374638, counted by A374518.
These are positions of strict rows in A375128, sums A374706, ranks A375400.
Partitions (or reversed partitions) of this type are counted by A375134.
For identical instead of distinct we have A375396, counted by A115029.
The complement is A375399, counted by A375404.
For maxima instead of minima we have A375402, counted by A375133.
The complement for maxima is A375403, counted by A375401.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A034296, A036785, A046660, A065200, A065201, A358836, A374632, A374761, A374767, A374768, A375136.

Select[Range[100],UnsameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 169, 171

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

An anti-run is a sequence with no adjacent equal terms.
The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
Note the prime factors can alternatively be taken in weakly decreasing order.

			The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs ((2),(2,3,5),(5)), with minima (2,2,5), so 300 is in the sequence.
The prime factors of 450 are {2,3,3,5,5}, with maximal anti-runs ((2,3),(3,5),(5)), with minima (2,3,5), so 450 is not in the sequence.
The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    20: {1,1,3}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    44: {1,1,5}
    45: {2,2,3}
    48: {1,1,1,1,2}

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The complement for compositions is A374638, counted by A374518.
A version for compositions is A374639, counted by A374678.
Positions of non-strict rows in A375128, sums A374706, ranks A375400.
For identical instead of strict we have A375397, counted by A375405.
The complement is A375398, counted by A375134.
The complement for maxima instead of minima is A375402, counted by A375133.
For maxima instead of minima we have A375403, counted by A375401.
Partitions (or reversed partitions) of this type are counted by A375404.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A034296, A046660, A065200, A065201, A115029, A279790, A374632, A374761, A375136, A375396.

Select[Range[100],!UnsameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 33, 48, 63, 88, 116, 157, 204, 272, 349, 456, 581, 749, 946, 1205, 1511, 1904, 2371, 2960, 3661, 4538, 5577, 6862, 8389, 10257, 12472, 15164, 18348, 22192, 26731, 32177, 38593, 46254, 55256, 65952, 78500, 93340, 110706

Offset: 0

Gus Wiseman, Aug 17 2024

nonn

An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.

			The partition y = (3,2,2,1) has maximal ant-runs ((3,2),(2,1)), with maxima (3,2), so y is not counted under a(8).
The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (2111)   (222)     (2221)     (332)
                       (11111)  (2211)    (4111)     (2222)
                                (3111)    (22111)    (3311)
                                (21111)   (31111)    (5111)
                                (111111)  (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

For identical instead of distinct we have A239955, ranks A073492.
The complement is counted by A375133, ranks A375402.
The complement for minima instead of maxima is A375134, ranks A375398.
These partitions have Heinz numbers A375403.
For minima instead of maxima we have A375404, ranks A375399.
The reverse for identical instead of distinct is A375405, ranks A375397.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A034296, A115029, A141199, A279790, A358830, A358836, A374632, A374761, A375136, A375396, A375400.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]

A375404 Number of integer partitions of n whose minima of maximal anti-runs are not all different.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 7, 9, 14, 19, 30, 38, 56, 73, 102, 133, 179, 231, 307, 392, 511, 647, 831, 1046, 1328, 1658, 2084, 2586, 3219, 3970, 4909, 6016, 7386, 9005, 10988, 13330, 16175, 19531, 23580, 28350, 34067, 40788, 48809, 58215, 69383, 82461, 97917, 115976

Offset: 0

Gus Wiseman, Aug 17 2024

nonn

An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

Also the number of reversed integer partitions of n such that the minima of maximal anti-runs are not all different.

			The a(0) = 0 through a(8) = 14 reversed partitions:
  .  .  (11)  (111)  (22)    (113)    (33)      (115)      (44)
                     (112)   (1112)   (114)     (223)      (116)
                     (1111)  (11111)  (222)     (1114)     (224)
                                      (1113)    (1123)     (1115)
                                      (1122)    (1222)     (1124)
                                      (11112)   (11113)    (1133)
                                      (111111)  (11122)    (2222)
                                                (111112)   (11114)
                                                (1111111)  (11123)
                                                           (11222)
                                                           (111113)
                                                           (111122)
                                                           (1111112)
                                                           (11111111)

The complement for maxima instead of minima is A375133, ranks A375402.
The complement is counted by A375134, ranks A375398.
These partitions are ranked by A375399.
For maxima instead of minima we have A375401, ranks A375403.
For identical instead of distinct we have A375405, ranks A375397.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A034296, A046660, A073492, A115029, A141199, A239955, A279790, A358830, A358836, A375136.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]

A375396 Numbers not divisible by the square of any prime factor except (possibly) the least. Hooklike numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

Also numbers k such that the minima of the maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are identical. Here, an anti-run is a sequence with no adjacent equal parts, and the minima of the maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each. Note the prime factors can alternatively be taken in weakly decreasing order.
The complement is a superset of A036785 = products of a squarefree number and a prime power.
The asymptotic density of this sequence is (1/zeta(2)) * (1 + Sum_{p prime} (1/(p^2-p)) / Product_{primes q <= p} (1 + 1/q)) = 0.884855661165... . - Amiram Eldar, Oct 26 2024

			The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs {{2},{2,3,5},{5}}, with minima (2,2,5), so 300 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The complement is a superset of A036785.
For maxima instead of minima we have A065200, counted by A034296.
The complement for maxima is A065201, counted by A239955.
Partitions of this type are counted by A115029.
A version for compositions is A374519, counted by A374517.
Also positions of identical rows in A375128, sums A374706, ranks A375400.
The complement is A375397, counted by A375405.
For distinct instead of identical minima we have A375398, counts A375134.
The complement for distinct minima is A375399, counted by A375404.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A011782 comps counts compositions.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
See the formula section for the relationships with A005117, A028234.
Cf. A000005, A013661, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.

Select[Range[100],SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) == e[1], 1); \\ Amiram Eldar, Oct 26 2024

{a(n)} = {k >= 1 : A028234(k) is in A005117}. - Peter Munn, May 09 2025

A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 6, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 12, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 10, 3, 4, 53, 18, 5, 8, 3, 2, 59, 4, 61, 2, 9, 64, 5, 2, 67, 4, 3, 2, 71, 24, 73, 2, 15, 4, 7

Offset: 1

Gus Wiseman, Aug 17 2024

nonn

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.
An anti-run is a sequence with no adjacent equal parts. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 540 are (1,1,2,2,2,3), with maximal anti-runs ((1),(1,2),(2),(2,3)), with minima (1,1,2,2), with Heinz number 36, so a(540) = 36.
The prime indices of 990 are (1,2,2,3,5), with maximal anti-runs ((1,2),(2,3,5)), with minima (1,2), with Heinz number 6, so a(990) = 6.

bigomega is A001222(a(n)) = A375136(n).
Least prime factor is A020639(a(n)) = A020639(n).
Least prime index is A055396(a(n)) = A055396(n).
Heinz weights are A056239(a(n)) = A374706(n).
The greatest prime index A061395(a(n)) is the maximum of row n of A375128.
Firsts for omega (except first term) are half A061742.
Prime indices A112798(a(n)) are row n of A375128.
Positions of prime-powers are A375396, counted by A115029.
Positions of squarefree numbers are A375398, counted by A375134.
A000041 counts integer partitions, strict A000009.
A027748 lists distinct prime factors, sum A008472.
A304038 lists distinct prime indices, sum A066328.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A000040, A046660, A124768, A320324, A358836, A374683, A374700, A375133.

Table[Times@@Prime/@If[n==1,{},Min /@ Split[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],UnsameQ]],{n,100}]

cp's OEIS Frontend

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

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A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.

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A375134 Number of integer partitions of n whose maximal anti-runs have distinct minima.

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A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

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A375398 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are distinct.

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A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.

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A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

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A375404 Number of integer partitions of n whose minima of maximal anti-runs are not all different.

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