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A324758
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Heinz numbers of integer partitions containing no prime indices of the parts.
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32
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1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
20: {1,1,3}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
27: {2,2,2}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Intersection[primeMS[#], Union@@primeMS/@primeMS[#]]=={}&]
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CROSSREFS
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The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.
Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.
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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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