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Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.
13

%I #7 Feb 18 2023 20:49:37

%S 4,10,15,22,24,25,33,34,36,40,46,51,54,55,56,62,69,77,82,85,88,93,94,

%T 100,104,115,118,119,121,123,134,135,136,141,146,152,155,161,166,177,

%U 184,187,194,196,201,205,206,217,218,219,220,221,225,232,235,240,248

%N Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

%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.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is in the sequence.

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

%t Select[Range[2,100],!IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

%Y For mean instead of median complement we have A340610, counted by A168659.

%Y For mean instead of median we have A360668, counted by A200727.

%Y Positions of odd terms in A360555.

%Y The complement is A360556 (without 1), counted by A360688.

%Y These partitions are counted by A360691.

%Y - For divisors (A063655) we have A139710, complement A139711.

%Y - For prime indices (A360005) we have A359912, complement A359908.

%Y - For distinct prime indices (A360457) we have A360551, complement A360550.

%Y - For distinct prime factors (A360458) we have A100367, complement A360552.

%Y - For prime factors (A360459) we have A072978, complement A359913.

%Y - For prime multiplicities (A360460) we have A360554, complement A360553.

%Y - For 0-prepended differences (A360555) we have A360557, complement A360556.

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

%Y A287352 lists 0-prepended first differences of prime indices.

%Y A325347 counts partitions with integer median, complement A307683.

%Y A355536 lists first differences of prime indices.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y A360614/A360615 = mean of first differences of 0-prepended prime indices.

%Y Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 17 2023