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A305732
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Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.
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12
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2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
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OFFSET
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1,1
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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). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1,...,m_k) = 1 and the multiset {y_1,...,y_k} is also reducible.
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LINKS
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EXAMPLE
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60 has relatively prime prime indices {1,1,2,3} with multiplicities {1,1,2} corresponding to A181819(90) = 12. 12 has relatively prime prime indices {1,1,2} with multiplicities {1,2} corresponding to A181819(12) = 6. 6 has relatively prime prime indices {1,2} with multiplicities {1,1} corresponding to A181819(6) = 4. 4 has relatively prime prime indices {1,1} with multiplicities {2} corresponding to A181819(4) = 3. 3 is prime, so we conclude that 60 belongs to the sequence.
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MATHEMATICA
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rdzQ[n_]:=And[n>1, Or[PrimeQ[n], And[rdzQ[Times@@Prime/@FactorInteger[n][[All, 2]]], GCD@@PrimePi/@FactorInteger[n][[All, 1]]==1]]];
Select[Range[50], rdzQ]
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CROSSREFS
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Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A304465, A304687, A304818, A305563, A305731, A305733.
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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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