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A325369
Numbers with no two prime exponents appearing the same number of times in the prime signature.
6
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86
OFFSET
1,2
COMMENTS
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities. The enumeration of these partitions by sum is given by A325329.
EXAMPLE
Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:
12: {1,1,2}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
50: {1,3,3}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
63: {2,2,4}
68: {1,1,7}
72: {1,1,1,2,2}
75: {2,3,3}
76: {1,1,8}
80: {1,1,1,1,3}
88: {1,1,1,5}
For example, the prime indices of 1260 are {1,1,2,2,3,4}, whose multiplicities give the prime signature {1,1,2,2}, and since 1 and 2 appear the same number of times, 1260 is not in the sequence.
MATHEMATICA
Select[Range[100], UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved