

A325328


Heinz numbers of finite arithmetic progressions (integer partitions with equal differences).


20



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 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
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OFFSET

1,2


COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A049988.


LINKS

Table of n, a(n) for n=1..67.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.


EXAMPLE

Most small numbers are in the sequence. However the sequence of nonterms 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}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
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}
68: {1,1,7}
70: {1,3,4}
For example, 60 is the Heinz number of (3,2,1,1), which has differences (1,1,0), which are not equal, so 60 does not belong to the sequence.


MATHEMATICA

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


CROSSREFS

Cf. A000961, A007862, A049988, A056239, A112798, A130091, A240026, A289509, A307824, A325327, A325352, A325368, A325405, A325407.
Sequence in context: A236510 A317710 A303554 * A316521 A085156 A102466
Adjacent sequences: A325325 A325326 A325327 * A325329 A325330 A325331


KEYWORD

nonn


AUTHOR

Gus Wiseman, Apr 23 2019


STATUS

approved



