A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91

Offset

1,2

Comments

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.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Links

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

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Example

The prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], Less@@Divide@@@Reverse/@Partition[primeptn[#], 2, 1]&]

Crossrefs

For differences instead of quotients we have A325456 (count: A240027).

For multiplicities (prime signature) instead of quotients we have A334965.

The version counting strict divisor chains is A342086.

These partitions are counted by A342498 (strict: A342517, ordered: A342493).

The weakly increasing version is A342523.

The strictly decreasing version is A342525.

A001055 counts factorizations (strict: A045778, ordered: A074206).

A003238 counts chains of divisors summing to n - 1 (strict: A122651).

A167865 counts strict chains of divisors > 1 summing to n.

A318991/A318992 rank reversed partitions with/without integer quotients.

A342098 counts (strict) partitions with all adjacent parts x > 2y.

Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

Sequence in context: A344414 A356438 A325456 * A328161 A003796 A032896

Adjacent sequences: A342521 A342522 A342523 * A342525 A342526 A342527

Keyword

nonn

Author

Gus Wiseman, Mar 23 2021

Status

approved