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Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.
16

%I #8 Aug 27 2022 21:30:27

%S 1,2,4,8,10,16,20,32,40,50,64,80,100,110,128,160,200,220,250,256,320,

%T 400,440,500,512,550,640,800,880,1000,1024,1100,1210,1250,1280,1600,

%U 1760,1870,2000,2048,2200,2420,2500,2560,2750,3200,3520,3740,4000,4096,4400

%N Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

%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 Also positions of first appearances of rows in A356226.

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 4: {1,1}

%e 8: {1,1,1}

%e 10: {1,3}

%e 16: {1,1,1,1}

%e 20: {1,1,3}

%e 32: {1,1,1,1,1}

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

%e 50: {1,3,3}

%e 64: {1,1,1,1,1,1}

%e 80: {1,1,1,1,3}

%e 100: {1,1,3,3}

%e 110: {1,3,5}

%e 128: {1,1,1,1,1,1,1}

%e 160: {1,1,1,1,1,3}

%e 200: {1,1,1,3,3}

%e 220: {1,1,3,5}

%e 250: {1,3,3,3}

%e 256: {1,1,1,1,1,1,1,1}

%e 320: {1,1,1,1,1,1,3}

%e 400: {1,1,1,1,3,3}

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

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t Select[Range[1000],normQ[(primeMS[#]+1)/2]&]

%Y The partitions with these Heinz numbers are counted by A053251.

%Y This is the odd restriction of A055932.

%Y A subset of A066208 (numbers with all odd prime indices).

%Y This is the sorted version of A356603.

%Y These are the positions of first appearances of rows in A356226. Other statistics are:

%Y - length: A287170, firsts A066205

%Y - minimum: A356227

%Y - maximum: A356228

%Y - bisected length: A356229

%Y - standard composition: A356230

%Y - Heinz number: A356231

%Y - positions of first appearances: A356232 (this sequence)

%Y A001221 counts distinct prime factors, with sum A001414.

%Y A001223 lists the prime gaps, reduced A028334.

%Y A003963 multiplies together the prime indices.

%Y A056239 adds up the prime indices, row sums of A112798.

%Y A073491 lists numbers with gapless prime indices, complement A073492.

%Y Cf. A000005, A001222, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 20 2022