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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 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
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[#]]]&]
CROSSREFS
Sequence in context: A263837 A283550 A271113 * A317101 A304449 A085924
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved

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Last modified July 25 01:41 EDT 2024. Contains 374585 sequences. (Running on oeis4.)