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A342524
Heinz numbers of integer partitions with strictly increasing first quotients.
5
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).
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.
Sequence in context: A344414 A356438 A325456 * A328161 A003796 A032896
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 23 2021
STATUS
approved