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A325118
Heinz numbers of binary carry-connected integer partitions.
16
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 19, 20, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 59, 60, 61, 62, 64, 65, 67, 68, 71, 73, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 94, 97, 100
OFFSET
1,2
COMMENTS
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 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.
EXAMPLE
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}
15: {2,3}
16: {1,1,1,1}
17: {7}
19: {8}
20: {1,1,3}
22: {1,5}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
MATHEMATICA
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], Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 28 2019
STATUS
approved