A390093 - OEIS

%I #7 Dec 07 2025 10:39:11

%S 2,4,5,8,10,11,16,17,20,22,23,25,30,31,32,34,40,41,44,46,47,50,55,59,

%T 60,62,64,66,67,68,73,80,82,83,85,88,90,92,94,97,100,102,103,109,110,

%U 115,118,120,121,124,125,127,128,132,134,136,137,138,146,149,150

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

%F Intersection of A340932 and A244991.

%e The terms together with their prime indices begin:

%e     2: {1}

%e     4: {1,1}

%e     5: {3}

%e     8: {1,1,1}

%e    10: {1,3}

%e    11: {5}

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

%e    17: {7}

%e    20: {1,1,3}

%e    22: {1,5}

%e    23: {9}

%e    25: {3,3}

%e    30: {1,2,3}

%e    31: {11}

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

%e    34: {1,7}

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

%e    41: {13}

%e    44: {1,1,5}

%e    46: {1,9}

%e    47: {15}

%e    50: {1,3,3}

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

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

%Y The case where all prime indices are odd is A066208.

%Y For just odd greatest prime index we have A244991, complement A244990.

%Y Partitions of this type are counted by A325338, strict A391228.

%Y For just odd least prime index we have A340932, complement A340933, counted by A026805.

%Y For sum instead of product we have A390988, counted by A390092, strict A390746.

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

%Y A001222 counts prime factors.

%Y A031368 lists odd-indexed primes.

%Y A055396 selects least prime index.

%Y A056239 adds up prime indices.

%Y A058695 counts partitions of odd numbers, ranks A300063.

%Y A061395 selects greatest prime index.

%Y A112798 lists the prime indices of each positive integer.

%Y Cf. A000009, A005408, A006530, A026424, A026804, A257991, A300272, A340604, A340854, A340855, A341446.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 05 2025