A390093 Numbers k > 1 such that the least prime index of k and the greatest prime index of k are both odd.

2, 4, 5, 8, 10, 11, 16, 17, 20, 22, 23, 25, 30, 31, 32, 34, 40, 41, 44, 46, 47, 50, 55, 59, 60, 62, 64, 66, 67, 68, 73, 80, 82, 83, 85, 88, 90, 92, 94, 97, 100, 102, 103, 109, 110, 115, 118, 120, 121, 124, 125, 127, 128, 132, 134, 136, 137, 138, 146, 149, 150

Offset

1,1

Comments

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..61.

Formula

Intersection of A340932 and A244991.

Example

The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    5: {3}
    8: {1,1,1}
   10: {1,3}
   11: {5}
   16: {1,1,1,1}
   17: {7}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}
   34: {1,7}
   40: {1,1,1,3}
   41: {13}
   44: {1,1,5}
   46: {1,9}
   47: {15}
   50: {1,3,3}

Mathematica

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

Crossrefs

The case where all prime indices are odd is A066208.

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

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

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

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

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

A001222 counts prime factors.

A031368 lists odd-indexed primes.

A055396 selects least prime index.

A056239 adds up prime indices.

A058695 counts partitions of odd numbers, ranks A300063.

A061395 selects greatest prime index.

A112798 lists the prime indices of each positive integer.

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

Sequence in context: A247886 A244991 A004612 * A066208 A388860 A357981

Adjacent sequences: A390090 A390091 A390092 * A390094 A390095 A390096

Keyword

nonn

Author

Gus Wiseman, Dec 05 2025

Status

approved