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A362605
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Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.
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32
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6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154
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OFFSET
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1,1
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COMMENTS
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A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
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LINKS
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EXAMPLE
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The prime indices of 180 are {1,1,2,2,3}, with modes {1,2}, so 180 is in the sequence, and the sequence differs from A182853.
The terms together with their prime indices begin:
6: {1,2}
10: {1,3}
14: {1,4}
15: {2,3}
21: {2,4}
22: {1,5}
26: {1,6}
30: {1,2,3}
33: {2,5}
34: {1,7}
35: {3,4}
36: {1,1,2,2}
38: {1,8}
39: {2,6}
42: {1,2,4}
46: {1,9}
51: {2,7}
55: {3,5}
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MAPLE
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q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):
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MATHEMATICA
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Select[Range[100], Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]
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PROG
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(Python)
from sympy import factorint
def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1
(PARI) is(n) = {my(e = factor(n)[, 2]); if(#e < 2, 0, e = vecsort(e); e[#e-1] == e[#e]); } \\ Amiram Eldar, Jan 20 2024
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CROSSREFS
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The first term with bigomega n appears to be A166023(n).
Partitions of this type are counted by A362607.
These are the positions of terms > 1 in A362611.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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