cp's OEIS Frontend

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

%I A352143 #8 Mar 18 2022 13:09:30
%S A352143 1,2,5,8,11,17,20,23,31,32,41,44,47,59,67,68,73,80,83,92,97,103,109,
%T A352143 124,125,127,128,137,149,157,164,167,176,179,188,191,197,211,227,233,
%U A352143 236,241,257,268,269,272,275,277,283,292,307,313,320,331,332,347,353
%N A352143 Numbers whose prime indices and conjugate prime indices are all odd.
%C A352143 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, sum A056239, length A001222.
%C A352143 A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
%C A352143 These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.
%F A352143 Intersection of A066208 and A346635.
%e A352143 The terms together with their prime indices begin:
%e A352143    1: {}
%e A352143    2: {1}
%e A352143    5: {3}
%e A352143    8: {1,1,1}
%e A352143   11: {5}
%e A352143   17: {7}
%e A352143   20: {1,1,3}
%e A352143   23: {9}
%e A352143   31: {11}
%e A352143   32: {1,1,1,1,1}
%e A352143   41: {13}
%e A352143   44: {1,1,5}
%e A352143   47: {15}
%e A352143   59: {17}
%e A352143   67: {19}
%e A352143   68: {1,1,7}
%e A352143   73: {21}
%e A352143   80: {1,1,1,1,3}
%t A352143 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352143 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352143 Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]
%Y A352143 The restriction to primes is A031368.
%Y A352143 These partitions appear to be counted by A053253.
%Y A352143 The even version is A066207^2.
%Y A352143 For even instead of odd conjugate parts we get A066208^2.
%Y A352143 The first condition alone (all odd indices) is A066208, counted by A000009.
%Y A352143 The second condition alone is A346635, counted by A000009.
%Y A352143 A055922 counts partitions with odd multiplicities, ranked by A268335.
%Y A352143 A066207 = indices all even, counted by A035363 (complement A086543).
%Y A352143 A109297 = same indices as exponents, counted by A114640.
%Y A352143 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A352143 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352143 A162642 counts odd prime exponents, even A162641.
%Y A352143 A238745 gives the Heinz number of the conjugate prime signature.
%Y A352143 A257991 counts odd indices, even A257992.
%Y A352143 A258116 ranks strict partitions with all odd parts, even A258117.
%Y A352143 A351979 = odd indices and even multiplicities, counted by A035457.
%Y A352143 A352140 = even indices and odd multiplicities, counted by A055922 aerated.
%Y A352143 A352141 = even indices and even multiplicities, counted by A035444.
%Y A352143 A352142 = odd indices and odd multiplicities, counted by A117958.
%Y A352143 Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.
%K A352143 nonn
%O A352143 1,2
%A A352143 _Gus Wiseman_, Mar 18 2022