A352143 - OEIS

%I #8 Mar 18 2022 13:09:30

%S 1,2,5,8,11,17,20,23,31,32,41,44,47,59,67,68,73,80,83,92,97,103,109,

%T 124,125,127,128,137,149,157,164,167,176,179,188,191,197,211,227,233,

%U 236,241,257,268,269,272,275,277,283,292,307,313,320,331,332,347,353

%N Numbers whose prime indices and conjugate prime indices are all odd.

%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, sum A056239, length A001222.

%C 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 These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

%F Intersection of A066208 and A346635.

%e The terms together with their prime indices begin:

%e    1: {}

%e    2: {1}

%e    5: {3}

%e    8: {1,1,1}

%e   11: {5}

%e   17: {7}

%e   20: {1,1,3}

%e   23: {9}

%e   31: {11}

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

%e   41: {13}

%e   44: {1,1,5}

%e   47: {15}

%e   59: {17}

%e   67: {19}

%e   68: {1,1,7}

%e   73: {21}

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

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

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

%Y The restriction to primes is A031368.

%Y These partitions appear to be counted by A053253.

%Y The even version is A066207^2.

%Y For even instead of odd conjugate parts we get A066208^2.

%Y The first condition alone (all odd indices) is A066208, counted by A000009.

%Y The second condition alone is A346635, counted by A000009.

%Y A055922 counts partitions with odd multiplicities, ranked by A268335.

%Y A066207 = indices all even, counted by A035363 (complement A086543).

%Y A109297 = same indices as exponents, counted by A114640.

%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

%Y A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

%Y A162642 counts odd prime exponents, even A162641.

%Y A238745 gives the Heinz number of the conjugate prime signature.

%Y A257991 counts odd indices, even A257992.

%Y A258116 ranks strict partitions with all odd parts, even A258117.

%Y A351979 = odd indices and even multiplicities, counted by A035457.

%Y A352140 = even indices and odd multiplicities, counted by A055922 aerated.

%Y A352141 = even indices and even multiplicities, counted by A035444.

%Y A352142 = odd indices and odd multiplicities, counted by A117958.

%Y Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 18 2022