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A344416
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Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.
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10
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3, 4, 7, 9, 10, 12, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 40, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146
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OFFSET
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1,1
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Also numbers m whose sum of prime indices A056239(m) is even and is at most twice the greatest prime index A061395(m).
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LINKS
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FORMULA
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EXAMPLE
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The sequence of terms together with their prime indices begins:
3: {2} 37: {12} 71: {20}
4: {1,1} 39: {2,6} 76: {1,1,8}
7: {4} 40: {1,1,1,3} 79: {22}
9: {2,2} 43: {14} 82: {1,13}
10: {1,3} 46: {1,9} 84: {1,1,2,4}
12: {1,1,2} 49: {4,4} 85: {3,7}
13: {6} 52: {1,1,6} 87: {2,10}
19: {8} 53: {16} 88: {1,1,1,5}
21: {2,4} 55: {3,5} 89: {24}
22: {1,5} 57: {2,8} 91: {4,6}
25: {3,3} 61: {18} 94: {1,15}
28: {1,1,4} 62: {1,11} 101: {26}
29: {10} 63: {2,2,4} 102: {1,2,7}
30: {1,2,3} 66: {1,2,5} 107: {28}
34: {1,7} 70: {1,3,4} 111: {2,12}
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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], EvenQ[Total[primeMS[#]]]&&Max[primeMS[#]]>=Total[primeMS[#]]/2&]
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CROSSREFS
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These partitions are counted by A000070 = even-indexed terms of A025065.
The opposite version with odd weights allowed appears to be A322109.
The conjugate opposite version allowing odds is A344291, counted by A110618.
A001222 counts prime factors with multiplicity.
A265640 lists Heinz numbers of palindromic partitions.
A301987 lists numbers whose sum of prime indices equals their product.
A334201 adds up all prime indices except the greatest.
A340387 lists Heinz numbers of partitions whose sum is twice their length.
Cf. A001414, A074761, A316413, A316428, A325037, A325038, A325044, A330950, A344293, A344294, A344297.
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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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