A360246 - OEIS

%I #8 Feb 08 2023 13:12:01

%S 12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,92,

%T 96,98,99,104,108,112,116,117,120,124,126,132,135,136,140,144,147,148,

%U 150,152,153,156,160,162,164,168,171,172,175,176,180,184,188,189

%N Numbers for which the prime indices do not have the same mean as the distinct prime indices.

%C First differs from A242416 in having 126.

%C Contains no squarefree numbers or perfect powers.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%e The terms together with their prime indices begin:

%e    12: {1,1,2}

%e    18: {1,2,2}

%e    20: {1,1,3}

%e    24: {1,1,1,2}

%e    28: {1,1,4}

%e    40: {1,1,1,3}

%e    44: {1,1,5}

%e    45: {2,2,3}

%e    48: {1,1,1,1,2}

%e    50: {1,3,3}

%e    52: {1,1,6}

%e    54: {1,2,2,2}

%e    56: {1,1,1,4}

%e    60: {1,1,2,3}

%e    63: {2,2,4}

%e    68: {1,1,7}

%e    72: {1,1,1,2,2}

%e The prime indices of 126 are {1,2,2,4} with mean 9/4 and distinct prime indices {1,2,4} with mean 7/3, so 126 is in the sequence.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Mean[prix[#]]!=Mean[Union[prix[#]]]&]

%Y Signature instead of parts: complement A324570, counted by A114638.

%Y Signature instead of distinct parts: complement A359903, counted by A360068.

%Y These partitions are counted by A360242.

%Y The complement is A360247, counted by A360243.

%Y For median we have A360248, counted by A360244 (complement A360245).

%Y Union of A360252 and A360253, counted by A360250 and A360251.

%Y A058398 counts partitions by mean, also A327482.

%Y A088529/A088530 gives mean of prime signature (A124010).

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

%Y A316413 = numbers whose prime indices have integer mean, distinct A326621.

%Y A326567/A326568 gives mean of prime indices.

%Y A326619/A326620 gives mean of distinct prime indices.

%Y Cf. A000975, A051293, A067340, A067538, A360005, A360241.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 07 2023