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Search: a317257 -id:a317257
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Heinz numbers of totally co-strong integer partitions.
+10
3
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
1,2
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.
EXAMPLE
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.
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[#]]]&]
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 04 2020
STATUS
approved
Smallest Heinz number of a superperiodic integer partition requiring n steps in the reduction to a multiset of size 1 obtained by repeatedly taking the multiset of multiplicities.
+10
2
2, 3, 9, 441, 11865091329, 284788749974468882877009302517495014698593896453070311184452244729
OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.
MATHEMATICA
Function[m, Times@@Prime/@m]/@NestList[Join@@Table[Table[2i, {Reverse[#][[i]]}], {i, Length[#]}]&, {1}, 4]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 12 2018
STATUS
approved
Heinz numbers of alternately strong integer partitions.
+10
0
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
1,2
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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[#]]]&]
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 09 2020
STATUS
approved

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