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A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.

%I A362605 #24 Jan 20 2024 03:10:39
%S A362605 6,10,14,15,21,22,26,30,33,34,35,36,38,39,42,46,51,55,57,58,62,65,66,
%T A362605 69,70,74,77,78,82,85,86,87,91,93,94,95,100,102,105,106,110,111,114,
%U A362605 115,118,119,122,123,129,130,133,134,138,141,142,143,145,146,154
%N A362605 Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.
%C A362605 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}.
%H A362605 Amiram Eldar, <a href="/A362605/b362605.txt">Table of n, a(n) for n = 1..10000</a>
%e A362605 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.
%e A362605 The terms together with their prime indices begin:
%e A362605      6: {1,2}
%e A362605     10: {1,3}
%e A362605     14: {1,4}
%e A362605     15: {2,3}
%e A362605     21: {2,4}
%e A362605     22: {1,5}
%e A362605     26: {1,6}
%e A362605     30: {1,2,3}
%e A362605     33: {2,5}
%e A362605     34: {1,7}
%e A362605     35: {3,4}
%e A362605     36: {1,1,2,2}
%e A362605     38: {1,8}
%e A362605     39: {2,6}
%e A362605     42: {1,2,4}
%e A362605     46: {1,9}
%e A362605     51: {2,7}
%e A362605     55: {3,5}
%p A362605 q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):
%p A362605 select(q, [$1..250])[];  # _Alois P. Heinz_, May 10 2023
%t A362605 Select[Range[100],Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]
%o A362605 (Python)
%o A362605 from sympy import factorint
%o A362605 def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1
%o A362605 print([k for k in range(155) if ok(k)]) # _Michael S. Branicky_, May 06 2023
%o A362605 (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
%Y A362605 The first term with bigomega n appears to be A166023(n).
%Y A362605 The complement is A356862, counted by A362608.
%Y A362605 For co-mode complement we have A359178, counted by A362610.
%Y A362605 For co-mode we have A362606, counted by A362609.
%Y A362605 Partitions of this type are counted by A362607.
%Y A362605 These are the positions of terms > 1 in A362611.
%Y A362605 A112798 lists prime indices, length A001222, sum A056239.
%Y A362605 A362614 counts partitions by number of modes, ranks A362611.
%Y A362605 A362615 counts partitions by number of co-modes, ranks A362613.
%Y A362605 Cf. A002865, A215366, A327473, A327476, A353864, A359908, A362612.
%K A362605 nonn
%O A362605 1,1
%A A362605 _Gus Wiseman_, May 05 2023