login
Numbers whose prime factorization has more than one mode. Numbers without a unique exponent of maximum frequency in the prime signature.
32

%I #24 Jan 20 2024 03:10:39

%S 6,10,14,15,21,22,26,30,33,34,35,36,38,39,42,46,51,55,57,58,62,65,66,

%T 69,70,74,77,78,82,85,86,87,91,93,94,95,100,102,105,106,110,111,114,

%U 115,118,119,122,123,129,130,133,134,138,141,142,143,145,146,154

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

%C 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 Amiram Eldar, <a href="/A362605/b362605.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 The terms together with their prime indices begin:

%e 6: {1,2}

%e 10: {1,3}

%e 14: {1,4}

%e 15: {2,3}

%e 21: {2,4}

%e 22: {1,5}

%e 26: {1,6}

%e 30: {1,2,3}

%e 33: {2,5}

%e 34: {1,7}

%e 35: {3,4}

%e 36: {1,1,2,2}

%e 38: {1,8}

%e 39: {2,6}

%e 42: {1,2,4}

%e 46: {1,9}

%e 51: {2,7}

%e 55: {3,5}

%p q:= n-> (l-> nops(l)>1 and l[-1]=l[-2])(sort(map(i-> i[2], ifactors(n)[2]))):

%p select(q, [$1..250])[]; # _Alois P. Heinz_, May 10 2023

%t Select[Range[100],Count[Last/@FactorInteger[#], Max@@Last/@FactorInteger[#]]>1&]

%o (Python)

%o from sympy import factorint

%o def ok(n): return n>1 and (e:=list(factorint(n).values())).count(max(e))>1

%o print([k for k in range(155) if ok(k)]) # _Michael S. Branicky_, May 06 2023

%o (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 The first term with bigomega n appears to be A166023(n).

%Y The complement is A356862, counted by A362608.

%Y For co-mode complement we have A359178, counted by A362610.

%Y For co-mode we have A362606, counted by A362609.

%Y Partitions of this type are counted by A362607.

%Y These are the positions of terms > 1 in A362611.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A362614 counts partitions by number of modes, ranks A362611.

%Y A362615 counts partitions by number of co-modes, ranks A362613.

%Y Cf. A002865, A215366, A327473, A327476, A353864, A359908, A362612.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 05 2023