A244989 -id:A244989 - cp's OEIS frontend

A244988 a(n) = n - A244989(n).

1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 6, 7, 8, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 21, 21, 22, 23, 23, 23, 23, 23, 24, 25, 25, 25, 26, 27, 28, 28, 29, 30, 31, 31, 31, 32, 32, 33, 33, 34, 34, 34, 34, 34, 35, 36

Offset: 1

Antti Karttunen, Jul 22 2014

nonn

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

Cf. A244989, A244990, A244321.

Scheme
```
(define  (A244988 n) (- n (A244989 n)))
```

a(n) = n - A244989(n).

For all n >= 1, a(A244990(n)) = n. [This sequence works as an inverse function for the injection A244990].

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

Original entry on oeis.org

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

Antti Karttunen, Jul 21 2014

nonn

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

			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)

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

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.

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

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

A244321 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = 2a(k), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = 1+(2a(k)).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 9, 12, 10, 13, 11, 15, 14, 16, 18, 17, 19, 24, 25, 20, 26, 21, 22, 27, 23, 31, 29, 30, 28, 32, 36, 34, 33, 37, 35, 39, 49, 38, 48, 51, 41, 50, 40, 52, 42, 53, 43, 44, 54, 45, 55, 47, 46, 63, 59, 61, 62, 58, 57, 60, 65, 56, 73, 64, 72, 68, 66, 69

Offset: 1

Antti Karttunen, Jul 22 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A244322.
Cf. A244988, A244989, A244990, A244991, A244992.
Similar entanglement permutations: A135141, A237427, A243287, A243343, A243345.

a(1) = 1, and for n > 1, if A244992(n) = 1 [i.e. the greatest prime factor of n has an odd index], a(n) = 2 * A244321(A244989(n)), otherwise, a(n) = 1 + (2 * A244321(A244988(n)-1)).

For all n >= 1, A000035(a(n)) = 1 - A244992(n).

A245614 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = A026424(a(k)), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = A028260(1+a(k)).

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 10, 5, 9, 12, 11, 16, 15, 24, 18, 8, 17, 14, 22, 20, 26, 19, 29, 25, 28, 36, 35, 55, 39, 44, 31, 13, 30, 27, 21, 38, 34, 51, 46, 42, 37, 57, 40, 47, 32, 52, 45, 62, 56, 50, 68, 60, 82, 81, 67, 121, 86, 93, 105, 72, 65, 79, 33, 59, 64, 23, 53, 48, 41, 58, 49, 85, 71, 77, 66, 111, 99

Offset: 1

Antti Karttunen, Jul 27 2014

This shares with the permutation A122111 the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245613.
Cf. A026424, A028260, A066829, A244988, A244989, A244990, A244991, A244992, A122111.
Similar permutations: A244321, A244322, A245604, A227413, A237126, A243288, A243344, A243346, A245606.

a(1) = 1, and for n > 1, if A244992(n) = 1, a(n) = A026424(a(A244989(n))), otherwise a(n) = A028260(1+a(A244988(n)-1)).
As a composition of related permutations:
a(n) = A245604(A244321(n)).
For all n >= 1, A244992(n) = A066829(a(n)).

cp's OEIS Frontend

A244988 a(n) = n - A244989(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A244321 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = 2a(k), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = 1+(2a(k)).

Views

Author

Keywords

Links

Crossrefs

Formula

A245614 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = A026424(a(k)), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = A028260(1+a(k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

cp's OEIS Frontend

A244988 a(n) = n - A244989(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A244321 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = 2*a(k), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = 1+(2*a(k)).

Views

Author

Keywords

Links

Crossrefs

Formula

A245614 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = A026424(a(k)), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = A028260(1+a(k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A244321 Permutation of natural numbers: a(1)=1; thereafter, if n is k-th number whose greatest prime factor has an odd index [i.e., n = A244991(k)], a(n) = 2a(k), otherwise, when n is k-th number whose greatest prime factor has an even index [i.e., n = A244990(1+k)], a(n) = 1+(2a(k)).