A342522 Heinz numbers of integer partitions with constant (equal) first quotients.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

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.

Wikipedia, Arithmetic progression

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

Example

The prime indices of 2093 are {4,6,9}, with first quotients (3/2,3/2), so 2093 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}

Mathematica

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

Crossrefs

For multiplicities (prime signature) instead of quotients we have A072774.

The version counting strict divisor chains is A169594.

For differences instead of quotients we have A325328 (count: A049988).

These partitions are counted by A342496 (strict: A342515, ordered: A342495).

The distinct instead of equal version is A342521.

A000005 count constant partitions.

A000041 counts partitions (strict: A000009).

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.

A342086 counts strict chains of divisors with strictly increasing quotients.

Cf. A003242, A047966, A049980, A056239, A067824, A112798, A118914, A124010, A130091, A181819, A325351, A325352.

Sequence in context: A325328 A316521 A085156 * A102466 A354144 A322547

Adjacent sequences: A342519 A342520 A342521 * A342523 A342524 A342525

Keyword

nonn

Author

Gus Wiseman, Mar 23 2021

Status

approved