A340932 Numbers whose least prime index is odd. Heinz numbers of integer partitions whose last part is odd.

2, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 54, 55, 56, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 80, 82, 83, 84, 85, 86, 88, 90, 92, 94, 95, 96, 97

Offset: 1

Author

Gus Wiseman, Feb 12 2021

Keywords

• nonn

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. 1 has no prime indices so is not included.

Examples

			The sequence of terms together with their prime indices begins:
      2: {1}           24: {1,1,1,2}       46: {1,9}
      4: {1,1}         25: {3,3}           47: {15}
      5: {3}           26: {1,6}           48: {1,1,1,1,2}
      6: {1,2}         28: {1,1,4}         50: {1,3,3}
      8: {1,1,1}       30: {1,2,3}         52: {1,1,6}
     10: {1,3}         31: {11}            54: {1,2,2,2}
     11: {5}           32: {1,1,1,1,1}     55: {3,5}
     12: {1,1,2}       34: {1,7}           56: {1,1,1,4}
     14: {1,4}         35: {3,4}           58: {1,10}
     16: {1,1,1,1}     36: {1,1,2,2}       59: {17}
     17: {7}           38: {1,8}           60: {1,1,2,3}
     18: {1,2,2}       40: {1,1,1,3}       62: {1,11}
     20: {1,1,3}       41: {13}            64: {1,1,1,1,1,1}
     22: {1,5}         42: {1,2,4}         65: {3,6}
     23: {9}           44: {1,1,5}         66: {1,2,5}
		

Crossrefs

These partitions are counted by A026804.

The case where all prime indices are odd is A066208.

Looking at greatest prime index instead of least gives A244991.

Every term x is a product of A257991(x) elements of A341446.

The complement is {1} \/ A340933, counted by A026805.

A001222 counts prime factors.

A005408 lists odd numbers.

A027193 counts odd-length partitions, ranked by A026424.

A031368 lists odd-indexed primes.

A055396 selects least prime index.

A056239 adds up prime indices.

A058695 counts partitions of odd numbers, ranked by A300063.

A061395 selects greatest prime index.

A112798 lists the prime indices of each positive integer.

Cf. A006141, A160786, A300272, A340604, A340854, A340855.

Programs

Mathematica

Select[Range[100],OddQ[PrimePi[FactorInteger[#][[1,1]]]]&]

Formula

A055396(a(n)) belongs to A005408.

Closed under multiplication.