A316529 -id:A316529 - cp's OEIS frontend

A316496 Number of totally strong integer partitions of n.

1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668

Offset: 0

Gus Wiseman, Jul 29 2018

nonn

An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.

			The a(1) = 1 through a(8) = 12 totally strong partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.

Cf. A181819, A182850, A182857, A304660, A305563, A316597.
The Heinz numbers of these partitions are A316529.
The version for compositions is A332274.
The dual version is A332275.
The version for reversed partitions is (also) A332275.
The narrowly normal version is A332297.
The alternating version is A332339 (see also A317256).
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.

totincQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]

Updated with corrected terminology by Gus Wiseman, Mar 07 2020

A329138 Numbers whose prime signature is a necklace.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A304678 in having 1350 = 2^1 * 3^3 * 5^2. First differs from A316529 in having 150 = 2^1 * 3^1 * 5^2.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   6: (1,1)
   7: (1)
   8: (3)
   9: (2)
  10: (1,1)
  11: (1)
  13: (1)
  14: (1,1)
  15: (1,1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  21: (1,1)
  22: (1,1)

Complement of A329142.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is aperiodic are A329139.
Cf. A001037, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329140.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],neckQ[Last/@FactorInteger[#]]&]

A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

The first term of this sequence that is absent from A112769 is 150.

An integer partition is totally nondecreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly increasing and, taken in reverse order, are themselves a totally nondecreasing integer partition.

			150 is the Heinz number of (3,3,2,1), with multiplicities (1,1,2), which has multiplicities (2,1), which are decreasing, so 150 does not belong to the sequence.

Complement of A316529.

Cf. A056239, A071365, A112769, A181819, A182850, A242031, A296150, A305733, A316496, A317258.

A335376 Heinz numbers of totally co-strong integer partitions.

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, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jun 04 2020

nonn

First differs from A242031 and A317257 in lacking 60.
A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
    1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
    2: {1}         17: {7}           33: {2,5}
    3: {2}         19: {8}           34: {1,7}
    4: {1,1}       20: {1,1,3}       35: {3,4}
    5: {3}         21: {2,4}         36: {1,1,2,2}
    6: {1,2}       22: {1,5}         37: {12}
    7: {4}         23: {9}           38: {1,8}
    8: {1,1,1}     24: {1,1,1,2}     39: {2,6}
    9: {2,2}       25: {3,3}         40: {1,1,1,3}
   10: {1,3}       26: {1,6}         41: {13}
   11: {5}         27: {2,2,2}       42: {1,2,4}
   12: {1,1,2}     28: {1,1,4}       43: {14}
   13: {6}         29: {10}          44: {1,1,5}
   14: {1,4}       30: {1,2,3}       45: {2,2,3}
   15: {2,3}       31: {11}          46: {1,9}
For example, 180 is the Heinz number of (3,2,2,1,1) which has run-lengths: (1,2,2) -> (1,2) -> (1,1) -> (2) -> (1). All of these are weakly increasing, so 180 is in the sequence.

Partitions with weakly increasing run-lengths are A100883.
Totally strong partitions are counted by A316496.
The strong version is A316529.
The version for reversed partitions is (also) A316529.
These partitions are counted by A332275.
The widely normal version is A332293.
The complement is A335377.
Cf. A100882, A133808, A181819, A182850, A242031, A305732, A317256, A317258, A329747, A332291.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];
Select[Range[100],totcostrQ[Reverse[primeMS[#]]]&]

A335377 Heinz numbers of non-totally co-strong integer partitions.

Original entry on oeis.org

18, 50, 54, 60, 75, 84, 90, 98, 108, 120, 126, 132, 140, 147, 150, 156, 162, 168, 198, 204, 220, 228, 234, 240, 242, 245, 250, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 324, 336, 338, 340, 342, 348, 350, 363, 364, 372, 375, 378, 380, 408, 414, 420

Offset: 1

Gus Wiseman, Jun 05 2020

nonn

A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.

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

			The sequence of terms together with their prime indices begins:
   18: {1,2,2}        156: {1,1,2,6}        276: {1,1,2,9}
   50: {1,3,3}        162: {1,2,2,2,2}      280: {1,1,1,3,4}
   54: {1,2,2,2}      168: {1,1,1,2,4}      294: {1,2,4,4}
   60: {1,1,2,3}      198: {1,2,2,5}        300: {1,1,2,3,3}
   75: {2,3,3}        204: {1,1,2,7}        306: {1,2,2,7}
   84: {1,1,2,4}      220: {1,1,3,5}        308: {1,1,4,5}
   90: {1,2,2,3}      228: {1,1,2,8}        312: {1,1,1,2,6}
   98: {1,4,4}        234: {1,2,2,6}        315: {2,2,3,4}
  108: {1,1,2,2,2}    240: {1,1,1,1,2,3}    324: {1,1,2,2,2,2}
  120: {1,1,1,2,3}    242: {1,5,5}          336: {1,1,1,1,2,4}
  126: {1,2,2,4}      245: {3,4,4}          338: {1,6,6}
  132: {1,1,2,5}      250: {1,3,3,3}        340: {1,1,3,7}
  140: {1,1,3,4}      260: {1,1,3,6}        342: {1,2,2,8}
  147: {2,4,4}        264: {1,1,1,2,5}      348: {1,1,2,10}
  150: {1,2,3,3}      270: {1,2,2,2,3}      350: {1,3,3,4}
For example, 60 is the Heinz number of (3,2,1,1), which has run-lengths: (1,1,2) -> (2,1) -> (1,1) -> (2) -> (1). Since (2,1) is not weakly increasing, 60 is in the sequence.

Partitions with weakly increasing run-lengths are counted by A100883.
Totally strong partitions are counted by A316496.
Heinz numbers of totally strong partitions are A316529.
The version for reversed partitions is A316597.
The strong version is (also) A316597.
The alternating version is A317258.
Totally co-strong partitions are counted by A332275.
The complement is A335376.
Cf. A100882, A181819, A182850, A242031, A305732, A317256, A329747, A332291, A332293, A332339.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];
Select[Range[100],!totcostrQ[Reverse[primeMS[#]]]&]

A334969 Heinz numbers of alternately strong integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Jun 09 2020

nonn

First differs from A304678 in lacking 450.
First differs from A316529 (the totally strong version) in having 150.
A sequence is alternately strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and, when reversed, are themselves an alternately strong sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence does not contain 450, the Heinz number of (3,3,2,2,1), because, while the multiplicities are weakly decreasing, their reverse (1,2,2) does not have weakly decreasing multiplicities.

The co-strong version is A317257.
The case of reversed partitions is (also) A317257.
The total version is A316529.
These partitions are counted by A332339.
Totally co-strong partitions are counted by A332275.
Alternately co-strong compositions are counted by A332338.
Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altstrQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],altstrQ[Reverse[Length/@Split[q]]]]];
Select[Range[100],altstrQ[Reverse[primeMS[#]]]&]

cp's OEIS Frontend

A316496 Number of totally strong integer partitions of n.

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A329138 Numbers whose prime signature is a necklace.

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A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

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A335376 Heinz numbers of totally co-strong integer partitions.

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A335377 Heinz numbers of non-totally co-strong integer partitions.

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A334969 Heinz numbers of alternately strong integer partitions.

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