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A325328
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Heinz numbers of finite arithmetic progressions (integer partitions with equal differences).
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26
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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
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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Most small numbers are in the sequence. However 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}
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.
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MATHEMATICA
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primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], SameQ@@Differences[primeptn[#]]&]
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CROSSREFS
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Cf. A000961, A007862, A049988, A056239, A112798, A130091, A240026, A289509, A307824, A325327, A325352, A325368, A325405, A325407.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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