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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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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

Views

Author

Gus Wiseman, Jun 04 2020

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Jun 05 2020

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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[#]]]&]
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