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A325100
Heinz numbers of strict integer partitions with no binary carries.
6
1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 41, 42, 43, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 74, 79, 83, 86, 89, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 122, 123, 127, 131, 133, 137, 139, 142, 149, 151, 157, 158, 159
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.
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 have no carries. 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}
5: {3}
6: {1,2}
7: {4}
11: {5}
13: {6}
14: {1,4}
17: {7}
19: {8}
21: {2,4}
23: {9}
26: {1,6}
29: {10}
31: {11}
33: {2,5}
35: {3,4}
37: {12}
38: {1,8}
41: {13}
42: {1,2,4}
MATHEMATICA
binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[100], SquareFreeQ[#]&&stableQ[PrimePi/@First/@FactorInteger[#], Intersection[binpos[#1], binpos[#2]]!={}&]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 28 2019
STATUS
approved