A391232 Numbers k > 1 such that the least prime index of k and the greatest prime index of k are both even.

3, 7, 9, 13, 19, 21, 27, 29, 37, 39, 43, 49, 53, 57, 61, 63, 71, 79, 81, 87, 89, 91, 101, 105, 107, 111, 113, 117, 129, 131, 133, 139, 147, 151, 159, 163, 169, 171, 173, 181, 183, 189, 193, 195, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 261, 263

Offset

1,1

Comments

First differs from A066207 in having 105.

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.

Links

Table of n, a(n) for n=1..57.

Example

The terms together with their prime indices begin:
    3: {2}
    7: {4}
    9: {2,2}
   13: {6}
   19: {8}
   21: {2,4}
   27: {2,2,2}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   49: {4,4}
   53: {16}
   57: {2,8}
   61: {18}
   63: {2,2,4}
   71: {20}
   79: {22}
   81: {2,2,2,2}
   87: {2,10}
   89: {24}
   91: {4,6}
  101: {26}
  105: {2,3,4}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], EvenQ[First[prix[#]]]&&EvenQ[Last[prix[#]]]&]

Crossrefs

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

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

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

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

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

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

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

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

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

A055396 and A061395 select least/greatest prime index.

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

A112798 list prime indices, sum A056239.

A257991 and A257992 count odd/even prime indices.

A366322 lists numbers with at least odd prime index.

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

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

Sequence in context: A225495 A111223 A266836 * A066207 A141544 A172407

Adjacent sequences: A391229 A391230 A391231 * A391233 A391234 A391235

Keyword

nonn

Author

Gus Wiseman, Dec 11 2025

Status

approved