cp's OEIS Frontend

A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

%I A356232 #8 Aug 27 2022 21:30:27
%S A356232 1,2,4,8,10,16,20,32,40,50,64,80,100,110,128,160,200,220,250,256,320,
%T A356232 400,440,500,512,550,640,800,880,1000,1024,1100,1210,1250,1280,1600,
%U A356232 1760,1870,2000,2048,2200,2420,2500,2560,2750,3200,3520,3740,4000,4096,4400
%N A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.
%C A356232 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 A356232 Also positions of first appearances of rows in A356226.
%e A356232 The terms together with their prime indices begin:
%e A356232       1: {}
%e A356232       2: {1}
%e A356232       4: {1,1}
%e A356232       8: {1,1,1}
%e A356232      10: {1,3}
%e A356232      16: {1,1,1,1}
%e A356232      20: {1,1,3}
%e A356232      32: {1,1,1,1,1}
%e A356232      40: {1,1,1,3}
%e A356232      50: {1,3,3}
%e A356232      64: {1,1,1,1,1,1}
%e A356232      80: {1,1,1,1,3}
%e A356232     100: {1,1,3,3}
%e A356232     110: {1,3,5}
%e A356232     128: {1,1,1,1,1,1,1}
%e A356232     160: {1,1,1,1,1,3}
%e A356232     200: {1,1,1,3,3}
%e A356232     220: {1,1,3,5}
%e A356232     250: {1,3,3,3}
%e A356232     256: {1,1,1,1,1,1,1,1}
%e A356232     320: {1,1,1,1,1,1,3}
%e A356232     400: {1,1,1,1,3,3}
%t A356232 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356232 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A356232 Select[Range[1000],normQ[(primeMS[#]+1)/2]&]
%Y A356232 The partitions with these Heinz numbers are counted by A053251.
%Y A356232 This is the odd restriction of A055932.
%Y A356232 A subset of A066208 (numbers with all odd prime indices).
%Y A356232 This is the sorted version of A356603.
%Y A356232 These are the positions of first appearances of rows in A356226. Other statistics are:
%Y A356232 - length: A287170, firsts A066205
%Y A356232 - minimum: A356227
%Y A356232 - maximum: A356228
%Y A356232 - bisected length: A356229
%Y A356232 - standard composition: A356230
%Y A356232 - Heinz number: A356231
%Y A356232 - positions of first appearances: A356232 (this sequence)
%Y A356232 A001221 counts distinct prime factors, with sum A001414.
%Y A356232 A001223 lists the prime gaps, reduced A028334.
%Y A356232 A003963 multiplies together the prime indices.
%Y A356232 A056239 adds up the prime indices, row sums of A112798.
%Y A356232 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356232 Cf. A000005, A001222, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.
%K A356232 nonn
%O A356232 1,2
%A A356232 _Gus Wiseman_, Aug 20 2022