A391232 - OEIS

%I #6 Dec 11 2025 22:57:59

%S 3,7,9,13,19,21,27,29,37,39,43,49,53,57,61,63,71,79,81,87,89,91,101,

%T 105,107,111,113,117,129,131,133,139,147,151,159,163,169,171,173,181,

%U 183,189,193,195,199,203,213,223,229,237,239,243,247,251,259,261,263

%N Numbers k > 1 such that the least prime index of k and the greatest prime index of k are both even.

%C First differs from A066207 in having 105.

%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.

%e The terms together with their prime indices begin:

%e     3: {2}

%e     7: {4}

%e     9: {2,2}

%e    13: {6}

%e    19: {8}

%e    21: {2,4}

%e    27: {2,2,2}

%e    29: {10}

%e    37: {12}

%e    39: {2,6}

%e    43: {14}

%e    49: {4,4}

%e    53: {16}

%e    57: {2,8}

%e    61: {18}

%e    63: {2,2,4}

%e    71: {20}

%e    79: {22}

%e    81: {2,2,2,2}

%e    87: {2,10}

%e    89: {24}

%e    91: {4,6}

%e   101: {26}

%e   105: {2,3,4}

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

%t Select[Range[2,100],EvenQ[First[prix[#]]]&&EvenQ[Last[prix[#]]]&]

%Y For even greatest part we have A244990, counted by A027187, strict A067661.

%Y For odd greatest part we have A244991, counted by A027193, strict A026837.

%Y Partitions of this type are counted by A325346, strict A391233.

%Y For odd least part we have A340932, counted by A026804, strict A026832.

%Y For even least part we have A340933, counted by A026805, strict A026833.

%Y For both parts odd we have A390093, counted by A325338, strict A391228.

%Y For odd sum we have A390988, counted by A390092, strict A390746.

%Y For even sum we have A391226, counted by A391225, strict A391227.

%Y The complement for both odd is A391229, counted by A391230, strict A391231.

%Y A055396 and A061395 select least/greatest prime index.

%Y A066207 and A066208 list numbers with all even/odd prime indices.

%Y A112798 list prime indices, sum A056239.

%Y A257991 and A257992 count odd/even prime indices.

%Y A366322 lists numbers with at least odd prime index.

%Y A390430 gives least prime index plus greatest prime index, difference A243055.

%Y Cf. A000720, A001222, A005408, A005843, A006530, A026424, A035363, A035457, A300061, A300272, A324929.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 11 2025