A384007 Heinz numbers of non Look-and-Say section-sum partitions.

10, 14, 15, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190

Offset

1,1

Comments

First differs from A383514 in lacking 1000.

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 integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

Links

Table of n, a(n) for n=1..58.

Example

The terms together with their prime indices begin:
    10: {1,3}    57: {2,8}      94: {1,15}
    14: {1,4}    58: {1,10}     95: {3,8}
    15: {2,3}    62: {1,11}    100: {1,1,3,3}
    22: {1,5}    65: {3,6}     106: {1,16}
    26: {1,6}    69: {2,9}     111: {2,12}
    33: {2,5}    74: {1,12}    115: {3,9}
    34: {1,7}    77: {4,5}     118: {1,17}
    35: {3,4}    82: {1,13}    119: {4,7}
    38: {1,8}    85: {3,7}     122: {1,18}
    39: {2,6}    86: {1,14}    123: {2,13}
    46: {1,9}    87: {2,10}    129: {2,14}
    51: {2,7}    91: {4,6}     130: {1,3,6}
    55: {3,5}    93: {2,11}    133: {4,8}

Mathematica

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]], UnsameQ@@Join@@#&];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], disjointFamilies[prix[#]]=={}&&disjointFamilies[conj[prix[#]]]!={}&]

Crossrefs

Ranking sequences are shown in parentheses below.

These partitions are counted by A383509.

Negating both properties gives (A383516).

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A098859 counts Wilf partitions (A130091), conjugate (A383512).

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).

A239455 counts section-sum partitions (A381432), complement A351293 (A381433).

A383508 counts partitions that are both Look-and-Say and section-sum (A383515).

A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).

A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).

Cf. A000720, A001223, A212166, A238745, A325368, A383514, A383520, A383531, A384006.

Sequence in context: A229271 A088711 A383514 * A154774 A162708 A330210

Adjacent sequences: A384004 A384005 A384006 * A384008 A384009 A384010

Keyword

nonn

Author

Gus Wiseman, May 19 2025

Status

approved