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A325097
Heinz numbers of integer partitions whose distinct parts have no binary carries.
11
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 56, 57, 58, 59, 61, 63, 64, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99, 101
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 numbers whose distinct prime indices have no binary carries.
EXAMPLE
Most small numbers are in the sequence, however the sequence of non-terms together with their prime indices begins:
10: {1,3}
15: {2,3}
20: {1,1,3}
22: {1,5}
30: {1,2,3}
34: {1,7}
39: {2,6}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
46: {1,9}
50: {1,3,3}
51: {2,7}
55: {3,5}
60: {1,1,2,3}
62: {1,11}
65: {3,6}
66: {1,2,5}
68: {1,1,7}
70: {1,3,4}
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], stableQ[PrimePi/@First/@FactorInteger[#], Intersection[binpos[#1], binpos[#2]]!={}&]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 27 2019
STATUS
approved