A317257 Heinz numbers of alternately co-strong integer partitions.

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, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset

1,2

Comments

The first term absent from this sequence but present in A242031 is 180.

A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..67.

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}

Mathematica

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

Crossrefs

Cf. A056239, A100883, A181819, A182850, A242031, A296150, A305732, A317246.

These partitions are counted by A317256.

The complement is A317258.

Totally co-strong partitions are counted by A332275.

Alternately co-strong compositions are counted by A332338.

Alternately co-strong reversed partitions are counted by A332339.

The total version is A335376.

Cf. A182857, A304660, A305563, A316496, A332292, A332340.

Sequence in context: A076084 A151764 A093618 * A242031 A335376 A109427

Adjacent sequences: A317254 A317255 A317256 * A317258 A317259 A317260

Keyword

nonn

Author

Gus Wiseman, Jul 25 2018

Extensions

Updated with corrected terminology by Gus Wiseman, Jun 04 2020

Status

approved