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A319241
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Heinz numbers of strict integer partitions of even numbers. Squarefree numbers whose prime indices sum to an even number.
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6
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1, 3, 7, 10, 13, 19, 21, 22, 29, 30, 34, 37, 39, 43, 46, 53, 55, 57, 61, 62, 66, 70, 71, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 118, 129, 130, 131, 133, 134, 138, 139, 146, 151, 154, 155, 159, 163, 165, 166, 173, 181, 183, 186, 187, 190, 193
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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).
For every odd squarefree number, s, exactly one of s and 2s is a term.
Closed under the commutative operation A350066(.,.).
Closed under the commutative operation A059897(.,.) forming a subgroup of the positive integers considered as a group under A059897. As subgroups, this sequence and A028982 are each a transversal of the other.
(End)
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LINKS
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FORMULA
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EXAMPLE
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30 is the Heinz number of (3,2,1), which is strict and has even weight, so 30 belongs to the sequence.
The sequence of all even-weight strict partitions begins: (), (2), (4), (3,1), (6), (8), (4,2), (5,1), (10), (3,2,1), (7,1), (12), (6,2), (14), (9,1), (16), (5,3), (8,2), (18), (11,1), (5,2,1), (4,3,1).
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MATHEMATICA
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Select[Range[100], And[SquareFreeQ[#], EvenQ[Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]]]&]
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PROG
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(PARI) isok(m) = issquarefree(m) && !(vecsum(apply(primepi, factor(m)[, 1])) % 2); \\ Michel Marcus, Feb 08 2022
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CROSSREFS
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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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