A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 23, 25, 30, 31, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124, 125, 127, 128

Offset

1,1

Comments

Equally, numbers n for which A061395(n) is odd.

A122111 maps each one of these numbers to a unique term of A026424 and vice versa.

If the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), these are the Heinz numbers of partitions whose greatest part is odd, counted by A027193. - Gus Wiseman, Feb 08 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001

Formula

For all n, A244989(a(n)) = n.

Example

From Gus Wiseman, Feb 08 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}           32: {1,1,1,1,1}     64: {1,1,1,1,1,1}
      4: {1,1}         33: {2,5}           66: {1,2,5}
      5: {3}           34: {1,7}           67: {19}
      8: {1,1,1}       40: {1,1,1,3}       68: {1,1,7}
     10: {1,3}         41: {13}            69: {2,9}
     11: {5}           44: {1,1,5}         73: {21}
     15: {2,3}         45: {2,2,3}         75: {2,3,3}
     16: {1,1,1,1}     46: {1,9}           77: {4,5}
     17: {7}           47: {15}            80: {1,1,1,1,3}
     20: {1,1,3}       50: {1,3,3}         82: {1,13}
     22: {1,5}         51: {2,7}           83: {23}
     23: {9}           55: {3,5}           85: {3,7}
     25: {3,3}         59: {17}            88: {1,1,1,5}
     30: {1,2,3}       60: {1,1,2,3}       90: {1,2,2,3}
     31: {11}          62: {1,11}          92: {1,1,9}
(End)

Mathematica

Select[Range[100], OddQ[PrimePi[FactorInteger[#][[-1, 1]]]]&] (* Gus Wiseman, Feb 08 2021 *)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A244991 (MATCHING-POS 1 1 (COMPOSE odd? A061395)))

Crossrefs

Complement: A244990.

Cf. A006530, A026424, A031368, A122111, A244321, A244322, A244989.

Looking at least instead of greatest prime index gives A026804.

The partitions with these Heinz numbers are counted by A027193.

The case where Omega is odd also is A340386.

A001222 counts prime factors.

A056239 adds up prime indices.

A300063 ranks partitions of odd numbers.

A061395 selects maximum prime index.

A066208 ranks partitions into odd parts.

A112798 lists the prime indices of each positive integer.

A340931 ranks odd-length partitions of odd numbers.

Cf. A000009, A058695, A072233, A160786, A300272, A340101, A340385, A340604.

Sequence in context: A289058 A102798 A247886 * A004612 A066208 A357981

Adjacent sequences: A244988 A244989 A244990 * A244992 A244993 A244994

Keyword

nonn

Author

Antti Karttunen, Jul 21 2014

Status

approved