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A325119
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Heinz numbers of binary carry-connected strict integer partitions.
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10
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1, 2, 3, 5, 7, 10, 11, 13, 15, 17, 19, 22, 23, 29, 30, 31, 34, 37, 39, 41, 43, 46, 47, 51, 53, 55, 59, 61, 62, 65, 67, 71, 73, 77, 79, 82, 83, 85, 87, 89, 91, 93, 94, 97, 101, 102, 103, 107, 109, 110, 113, 115, 118, 119, 127, 129, 130, 131, 134, 137, 139, 141
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OFFSET
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1,2
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COMMENTS
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A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are binary carry-connected. 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.
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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}
5: {3}
7: {4}
10: {1,3}
11: {5}
13: {6}
15: {2,3}
17: {7}
19: {8}
22: {1,5}
23: {9}
29: {10}
30: {1,2,3}
31: {11}
34: {1,7}
37: {12}
39: {2,6}
41: {13}
43: {14}
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MATHEMATICA
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binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Select[Range[100], SquareFreeQ[#]&&Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]
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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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