A257992 -id:A257992 - cp's OEIS frontend

A366533 Sum of even prime indices of n divided by 2.

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 3, 2, 1, 0, 0, 2, 4, 0, 3, 0, 0, 1, 0, 3, 3, 2, 5, 1, 0, 0, 1, 0, 2, 2, 6, 4, 4, 0, 0, 3, 7, 0, 2, 0, 0, 1, 4, 0, 1, 3, 8, 3, 0, 2, 5, 5, 0, 1, 9, 0, 4, 0, 3, 1, 0, 0, 1, 2, 10, 2, 0, 6, 1, 4, 2, 4, 11, 0, 4, 0, 0, 3, 0, 7

Offset: 1

Gus Wiseman, Oct 23 2023

nonn

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.

			The prime indices of 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Zeros are A066208, counted by A000009.
The triangle for this statistic (without zeros) is A174713.
The un-halved odd version is A366528.
The un-halved version is A366531.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A365067 counts partitions by sum of odd parts (without zeros).
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

a(n) = A366531(n)/2.

A379312 Positive integers whose prime indices include a unique 1 or prime number.

Original entry on oeis.org

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A379316 Positive integers whose prime indices include a unique squarefree number.

Original entry on oeis.org

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211

Offset: 1

Gus Wiseman, Dec 29 2024

nonn

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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

For all squarefree parts we have A302478, zeros of A379310.
Positions of 1 in A379306.
For no squarefree parts we have A379307, counted by A114374, strict A256012.
Partitions of this type are counted by A379308, strict A379309.
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 prime or 1, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]

A324966 Number of distinct odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 0

Offset: 1

Gus Wiseman, Mar 21 2019

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.

If x and y are coprime then a(x*y) = a(x)+a(y). - Robert Israel, Mar 24 2019

			180180 has prime indices {1,1,2,2,3,4,5,6}, so a(180180) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A003963, A005087, A066208, A112798, A257991, A257992, A289509, A324929, A324967.

f:= proc(n) nops(select(type,map(numtheory:-pi,numtheory:-factorset(n)),odd)) end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019

Table[Count[If[n==1,{},FactorInteger[n]],{?(OddQ[PrimePi[#]]&),}],{n,100}]

a(n) = my(f=factor(n)[,1]); sum(k=1, #f, primepi(f[k]) % 2); \\ Michel Marcus, Mar 22 2019

a(n) = A001221(n) - A324967(n). - Robert Israel, Mar 24 2019
G.f.: Sum_{k>=1} x^prime(2*k-1) / (1 - x^prime(2*k-1)). - Ilya Gutkovskiy, Feb 12 2020
Additive with a(p^e) = 1 if primepi(p) is odd and 0 otherwise. - Amiram Eldar, Oct 06 2023

A349158 Heinz numbers of integer partitions with exactly one odd part.

Original entry on oeis.org

2, 5, 6, 11, 14, 15, 17, 18, 23, 26, 31, 33, 35, 38, 41, 42, 45, 47, 51, 54, 58, 59, 65, 67, 69, 73, 74, 77, 78, 83, 86, 93, 95, 97, 98, 99, 103, 105, 106, 109, 114, 119, 122, 123, 126, 127, 135, 137, 141, 142, 143, 145, 149, 153, 157, 158, 161, 162, 167, 174

Offset: 1

Gus Wiseman, Nov 12 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with exactly one odd prime index. These are also partitions whose conjugate partition has alternating sum equal to 1.

Numbers that are product of a term of A031368 and a term of A066207. - Antti Karttunen, Nov 13 2021

			The terms and corresponding partitions begin:
      2: (1)         42: (4,2,1)       86: (14,1)
      5: (3)         45: (3,2,2)       93: (11,2)
      6: (2,1)       47: (15)          95: (8,3)
     11: (5)         51: (7,2)         97: (25)
     14: (4,1)       54: (2,2,2,1)     98: (4,4,1)
     15: (3,2)       58: (10,1)        99: (5,2,2)
     17: (7)         59: (17)         103: (27)
     18: (2,2,1)     65: (6,3)        105: (4,3,2)
     23: (9)         67: (19)         106: (16,1)
     26: (6,1)       69: (9,2)        109: (29)
     31: (11)        73: (21)         114: (8,2,1)
     33: (5,2)       74: (12,1)       119: (7,4)
     35: (4,3)       77: (5,4)        122: (18,1)
     38: (8,1)       78: (6,2,1)      123: (13,2)
     41: (13)        83: (23)         126: (4,2,2,1)

These partitions are counted by A000070 up to 0's.
Allowing no odd parts gives A066207, counted by A000041 up to 0's.
Requiring all odd parts gives A066208, counted by A000009.
These are the positions of 1's in A257991.
The even prime indices are counted by A257992.
The conjugate partitions are ranked by A345958.
Allowing at most one odd part gives A349150, counted by A100824.
A047993 ranks balanced partitions, counted by A106529.
A056239 adds up prime indices, row sums of A112798.
A122111 is a representation of partition conjugation.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A340604 ranks partitions of odd positive rank, counted by A101707.
A340932 ranks partitions whose least part is odd, counted by A026804.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000700, A001222, A027187, A027193, A028260, A031368 (primes with odd index), A035363, A215366, A277579, A300063, A349151.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],_?OddQ]==1&]

A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      13         15         18         20           22
   ------------------------------------------------------------------
    (2,1)  (6,5,2)    (10,5)     (12,6)     (12,7,1)     (12,8,2)
           (6,4,2,1)  (6,4,3,2)  (8,7,3)    (8,5,4,3)    (8,6,5,3)
                      (6,5,3,1)  (8,5,3,2)  (8,6,4,2)    (8,7,5,2)
                                 (8,6,3,1)  (8,7,4,1)    (12,7,2,1)
                                            (8,6,3,2,1)  (8,6,4,3,1)
                                                         (8,7,4,2,1)

This is the strict case of A045931, ranked by A350848 (zeros of A350941).
The conjugate version is A239241, non-strict A045931 (ranked by A325698).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, ranked by A349157, strict A352131.
- A277103, ranked by A350944.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A379300 Number of prime indices of n that are composite.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

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.

			The prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

A379311 Number of prime indices of n that are 1 or prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 1, 3, 0, 1, 2, 4, 1, 3, 0, 3, 1, 2, 0, 4, 2, 1, 3, 2, 0, 3, 1, 5, 2, 2, 1, 4, 0, 1, 1, 4, 1, 2, 0, 3, 3, 1, 0, 5, 0, 3, 2, 2, 0, 4, 2, 3, 1, 1, 1, 4, 0, 2, 2, 6, 1, 3, 1, 3, 1, 2, 0, 5, 0, 1, 3, 2, 1, 2, 0, 5, 4, 2, 1, 3, 2, 1, 1

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

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.

			The prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 2.
The prime indices of 98 are {1,4,4}, so a(98) = 1.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 1.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Positions of first appearances are A000079.
These "old" primes are listed by A008578.
Positions of zero are A320629, counted by A023895 (strict A204389).
Positions of one are A379312, counted by A379314 (strict A379315).
Positions of nonzero terms are A379313.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526, A173390, A376683, A376855.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A034891, A036234, A036497, A038550, A087436, A302540, A379300, A379301.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],#==1||PrimeQ[#]&]],{n,100}]

Totally additive with a(prime(k)) = A080339(k).

A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Oct 14 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    5: {3}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}

The complement is A066207, counted by A035363.
For all odd parts we have A066208, counted by A000009.
Partitions of this type are counted by A086543.
For even instead of odd we have A324929, counted by A047967.
A031368 lists primes of odd index.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A257992, A318400, A324927, A358137.

Select[Range[100],Or@@OddQ/@PrimePi/@First/@FactorInteger[#]&]

A257991(a(n)) > 0.

A379307 Positive integers whose prime indices include no squarefree numbers.

Original entry on oeis.org

1, 7, 19, 23, 37, 49, 53, 61, 71, 89, 97, 103, 107, 131, 133, 151, 161, 173, 193, 197, 223, 227, 229, 239, 251, 259, 263, 281, 307, 311, 337, 343, 359, 361, 371, 379, 383, 409, 419, 427, 433, 437, 457, 463, 479, 497, 503, 521, 523, 529, 541, 569, 593, 613, 623

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

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.

			The terms together with their prime indices begin:
    1: {}
    7: {4}
   19: {8}
   23: {9}
   37: {12}
   49: {4,4}
   53: {16}
   61: {18}
   71: {20}
   89: {24}
   97: {25}
  103: {27}
  107: {28}
  131: {32}
  133: {4,8}
  151: {36}
  161: {4,9}
  173: {40}

Partitions of this type are counted by A114374, strict A256012.
Positions of zero in A379306.
For a unique squarefree part we have A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A072284, A087436, A112929, A377430, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==0&]

cp's OEIS Frontend

A366533 Sum of even prime indices of n divided by 2.

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A379312 Positive integers whose prime indices include a unique 1 or prime number.

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A379316 Positive integers whose prime indices include a unique squarefree number.

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A324966 Number of distinct odd prime indices of n.

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A349158 Heinz numbers of integer partitions with exactly one odd part.

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A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

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A379300 Number of prime indices of n that are composite.

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A379311 Number of prime indices of n that are 1 or prime.

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