A160786 -id:A160786 - cp's OEIS frontend

A340832 Number of factorizations of n into factors > 1 with odd least factor.

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)     (3*36)     (135)      (3*60)     (3*84)
  (5*9)    (9*12)     (3*45)     (5*36)     (7*36)
  (3*15)   (3*4*9)    (5*27)     (9*20)     (9*28)
  (3*3*5)  (3*6*6)    (9*15)     (5*6*6)    (3*3*28)
           (3*3*12)   (3*5*9)    (3*3*20)   (3*4*21)
           (3*3*3*4)  (3*3*15)   (3*4*15)   (3*6*14)
                      (3*3*3*5)  (3*5*12)   (3*7*12)
                                 (3*6*10)   (3*3*4*7)
                                 (3*3*4*5)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A340854.
Positions of nonzero terms are A340855.
The version for partitions is A026804.
Odd-length factorizations are counted by A339890.
The version looking at greatest factor is A340831.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340607 counts factorizations with odd length and greatest factor.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A026424 lists numbers with odd Omega.
A027193 counts partitions of odd length.
A058695 counts partitions of odd numbers (A300063).
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A244991 lists numbers whose greatest prime index is odd.
A340692 counts partitions of odd rank.
Cf. A050320, A160786, A340385, A340596, A340599, A340654, A340655, A340931.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ@*Min]],{n,100}]

A340832(n, m=n, fc=1) = if(1==n, (m%2)&&!fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A340832(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

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

Original entry on oeis.org

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

Gus Wiseman, Feb 12 2021

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. 1 has no prime indices so is not included.

			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}

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.

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

A055396(a(n)) belongs to A005408.

Closed under multiplication.

A340603 Heinz numbers of integer partitions of odd rank.

Original entry on oeis.org

3, 4, 7, 10, 12, 13, 15, 16, 18, 19, 22, 25, 27, 28, 29, 33, 34, 37, 40, 42, 43, 46, 48, 51, 52, 53, 55, 60, 61, 62, 63, 64, 69, 70, 71, 72, 76, 77, 78, 79, 82, 85, 88, 89, 90, 93, 94, 98, 100, 101, 105, 107, 108, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      3: (2)           33: (5,2)           63: (4,2,2)
      4: (1,1)         34: (7,1)           64: (1,1,1,1,1,1)
      7: (4)           37: (12)            69: (9,2)
     10: (3,1)         40: (3,1,1,1)       70: (4,3,1)
     12: (2,1,1)       42: (4,2,1)         71: (20)
     13: (6)           43: (14)            72: (2,2,1,1,1)
     15: (3,2)         46: (9,1)           76: (8,1,1)
     16: (1,1,1,1)     48: (2,1,1,1,1)     77: (5,4)
     18: (2,2,1)       51: (7,2)           78: (6,2,1)
     19: (8)           52: (6,1,1)         79: (22)
     22: (5,1)         53: (16)            82: (13,1)
     25: (3,3)         55: (5,3)           85: (7,3)
     27: (2,2,2)       60: (3,2,1,1)       88: (5,1,1,1)
     28: (4,1,1)       61: (18)            89: (24)
     29: (10)          62: (11,1)          90: (3,2,2,1)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A340692.
The complement is A340602, counted by A340601.
The case of positive rank is A340604.
- Rank -
A001222 gives number of prime indices.
A047993 counts partitions of rank 0 (A106529).
A061395 gives maximum prime index.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340608, A340609, A340610.

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

A061395(a(n)) - A001222(a(n)) is odd.

A340931 Heinz numbers of integer partitions of odd numbers into an odd number of parts.

Original entry on oeis.org

2, 5, 8, 11, 17, 18, 20, 23, 31, 32, 41, 42, 44, 45, 47, 50, 59, 67, 68, 72, 73, 78, 80, 83, 92, 97, 98, 99, 103, 105, 109, 110, 114, 124, 125, 127, 128, 137, 149, 153, 157, 162, 164, 167, 168, 170, 174, 176, 179, 180, 182, 188, 191, 195, 197, 200, 207, 211

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

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

			The sequence of terms together with the corresponding partitions begins:
      2: (1)             50: (3,3,1)        109: (29)
      5: (3)             59: (17)           110: (5,3,1)
      8: (1,1,1)         67: (19)           114: (8,2,1)
     11: (5)             68: (7,1,1)        124: (11,1,1)
     17: (7)             72: (2,2,1,1,1)    125: (3,3,3)
     18: (2,2,1)         73: (21)           127: (31)
     20: (3,1,1)         78: (6,2,1)        128: (1,1,1,1,1,1,1)
     23: (9)             80: (3,1,1,1,1)    137: (33)
     31: (11)            83: (23)           149: (35)
     32: (1,1,1,1,1)     92: (9,1,1)        153: (7,2,2)
     41: (13)            97: (25)           157: (37)
     42: (4,2,1)         98: (4,4,1)        162: (2,2,2,2,1)
     44: (5,1,1)         99: (5,2,2)        164: (13,1,1)
     45: (3,2,2)        103: (27)           167: (39)
     47: (15)           105: (4,3,2)        168: (4,2,1,1,1)

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A160786.
The even version is A236913 (A340784).
The case of where the prime indices are also odd is A300272.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts odd-length partitions (A026424).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
Cf. A000041, A316413, A326845, A340385, A340386, A340387, A340604, A340607, A340854, A340855.

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

Intersection of A026424 and A300063.

A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

Original entry on oeis.org

0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004

Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

			From _Gus Wiseman_, Jan 06 2021: (Start)
a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
  {{1}}  {{1,2}}  {{1,2,3}}
                  {{1},{2},{3}}
                  {{1},{3},{2}}
                  {{2},{1},{3}}
                  {{2},{3},{1}}
                  {{3},{1},{2}}
                  {{3},{2},{1}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013
Gottfried Helms, Discussion of a problem concerning summing of like powers

Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A058695 counts partitions of odd numbers, ranked by A300063.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.

h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # Peter Luschny, Jul 09 2015

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^(2*m+1)/prod(k=1,2*m+1,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A089677_list(len):  # with a(0)=1
    e, r = [1], [1]
    for i in (1..len-1):
        for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
        r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
        e.append(sum(e))
    return r
A089677_list(21) # Peter Luschny, Jul 09 2015

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005
a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015

A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

Original entry on oeis.org

2, 6, 7, 8, 10, 11, 15, 18, 19, 21, 24, 26, 27, 28, 29, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 65, 70, 71, 72, 74, 76, 78, 79, 81, 84, 86, 87, 89, 91, 95, 96, 98, 101, 104, 105, 106, 107, 108, 111, 112, 113, 114, 116, 117, 122, 126, 128

Offset: 1

Gus Wiseman, May 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 even version is A372589.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {2}   2  (1)
      {2,3}   6  (2,1)
    {1,2,3}   7  (4)
        {4}   8  (1,1,1)
      {2,4}  10  (3,1)
    {1,2,4}  11  (5)
  {1,2,3,4}  15  (3,2)
      {2,5}  18  (2,2,1)
    {1,2,5}  19  (8)
    {1,3,5}  21  (4,2)
      {4,5}  24  (2,1,1,1)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
    {3,4,5}  28  (4,1,1)
  {1,3,4,5}  29  (10)
        {6}  32  (1,1,1,1,1)
      {1,6}  33  (5,2)
      {2,6}  34  (7,1)
      {4,6}  40  (3,1,1,1)
    {1,4,6}  41  (13)
    {3,4,6}  44  (5,1,1)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
Positions of odd terms in A372442, zeros A372436.
The complement is A372589.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066208, A160786, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],OddQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Numbers k such that A070939(k) + A061395(k) is odd.

A340786 Number of factorizations of 4n into an even number of even factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Jan 31 2021

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
  4*6   6*8      2*48      2*72      4*60      4*80          40*50
  2*12  2*24     4*24      4*36      6*40      8*40          4*500
        4*12     6*16      6*24      8*30      10*32         8*250
        2*2*2*6  8*12      8*18      10*24     16*20         10*200
                 2*2*4*6   12*12     12*20     2*160         20*100
                 2*2*2*12  2*2*6*6   2*120     2*2*2*40      2*1000
                           2*2*2*18  2*2*2*30  2*2*4*20      2*2*10*50
                                     2*2*6*10  2*2*8*10      2*2*2*250
                                               2*4*4*10      2*10*10*10
                                               2*2*2*2*2*10

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

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of ones are 1 and A000040, or A008578.
A version for partitions is A027187 (A028260).
Allowing odd length gives A108501 (bisection of A340785).
Allowing odd factors gives A339846.
An odd version is A340102.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
- Even -
A027187 counts partitions of even maximum (A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

g:= proc(n, m, p)
 option remember;
 local F,r,x,i;
 # number of factorizations of n into even factors > m with number of factors == p (mod 2)
 if n = 1 then if p = 0 then return 1 else return 0 fi fi;
 if m > n  or n::odd then return 0 fi;
 F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)),list));
 r:= 0;
 for x in F do
   for i from 1 while n mod x^i = 0 do
     r:= r + procname(n/x^i, x, (p+i) mod 2)
 od od;
 r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n],EvenQ[Length[#]]&&Select[#,OddQ]=={}&]],{n,100}]

A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022

A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 15, 16, 17, 20, 21, 29, 32, 36, 42, 43, 45, 46, 47, 48, 51, 53, 54, 55, 59, 60, 61, 63, 64, 65, 66, 67, 68, 71, 73, 78, 79, 80, 81, 84, 89, 91, 93, 94, 95, 97, 99, 101, 105, 110, 111, 113, 114, 115, 116, 118, 119, 121, 122, 125, 127

Offset: 1

Gus Wiseman, May 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 even version is A372587.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {1}   1  ()
            {2}   2  (1)
          {1,2}   3  (2)
            {3}   4  (1,1)
          {1,3}   5  (3)
            {4}   8  (1,1,1)
          {1,4}   9  (2,2)
          {3,4}  12  (2,1,1)
      {1,2,3,4}  15  (3,2)
            {5}  16  (1,1,1,1)
          {1,5}  17  (7)
          {3,5}  20  (3,1,1)
        {1,3,5}  21  (4,2)
      {1,3,4,5}  29  (10)
            {6}  32  (1,1,1,1,1)
          {3,6}  36  (2,2,1,1)
        {2,4,6}  42  (4,2,1)
      {1,2,4,6}  43  (14)
      {1,3,4,6}  45  (3,2,2)
      {2,3,4,6}  46  (9,1)
    {1,2,3,4,6}  47  (15)
          {5,6}  48  (2,1,1,1,1)

Positions of odd terms in A372428, zeros A372427.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372587.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],OddQ[Total[bix[#]]+Total[prix[#]]]&]

Numbers k such that A029931(k) + A056239(k) is odd.

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

Original entry on oeis.org

1, 3, 4, 5, 12, 14, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 35, 38, 43, 45, 48, 49, 53, 55, 56, 62, 63, 64, 66, 68, 69, 71, 72, 74, 75, 78, 80, 81, 82, 83, 84, 87, 88, 89, 91, 92, 93, 94, 99, 100, 101, 102, 104, 105, 108, 113, 114, 115, 116, 118, 120

Offset: 1

Gus Wiseman, May 14 2024

nonn

The even version is A372591.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1}   1  ()
      {1,2}   3  (2)
        {3}   4  (1,1)
      {1,3}   5  (3)
      {3,4}  12  (2,1,1)
    {2,3,4}  14  (4,1)
        {5}  16  (1,1,1,1)
      {1,5}  17  (7)
      {2,5}  18  (2,2,1)
      {3,5}  20  (3,1,1)
    {1,3,5}  21  (4,2)
    {2,3,5}  22  (5,1)
  {1,2,3,5}  23  (9)
    {1,4,5}  25  (3,3)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
  {1,3,4,5}  29  (10)
  {2,3,4,5}  30  (3,2,1)
    {1,2,6}  35  (4,3)
    {2,3,6}  38  (8,1)
  {1,2,4,6}  43  (14)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586, complement A372587.
For minimum (A372437) we have A372439, complement A372440.
Positions of odd terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372591.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],OddQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A341447 Heinz numbers of integer partitions whose only even part is the smallest.

Original entry on oeis.org

3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317

Offset: 1

Gus Wiseman, Feb 13 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only even prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      3: (2)         77: (5,4)     165: (5,3,2)
      7: (4)         79: (22)      173: (40)
     13: (6)         89: (24)      177: (17,2)
     15: (3,2)       93: (11,2)    181: (42)
     19: (8)        101: (26)      193: (44)
     29: (10)       107: (28)      199: (46)
     33: (5,2)      113: (30)      201: (19,2)
     37: (12)       119: (7,4)     217: (11,4)
     43: (14)       123: (13,2)    219: (21,2)
     51: (7,2)      131: (32)      221: (7,6)
     53: (16)       139: (34)      223: (48)
     61: (18)       141: (15,2)    229: (50)
     69: (9,2)      151: (36)      239: (52)
     71: (20)       161: (9,4)     249: (23,2)
     75: (3,3,2)    163: (38)      251: (54)

These partitions are counted by A087897, shifted left once.
Terms of A340933 can be factored into elements of this sequence.
The odd version is A341446.
A000009 counts partitions into odd parts, ranked by A066208.
A001222 counts prime factors.
A005843 lists even numbers.
A026804 counts partitions whose least part is odd, ranked by A340932.
A026805 counts partitions whose least part is even, ranked by A340933.
A027187 counts partitions with even length/max, ranked by A028260/A244990.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058696 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A066207 lists numbers with all even prime indices.
A112798 lists the prime indices of each positive integer.
Cf. A026804, A035363, A036554, A160786, A244991, A257991, A257992, A300272, A300063, A340854/A340855.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]

cp's OEIS Frontend

A340832 Number of factorizations of n into factors > 1 with odd least factor.

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

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A340603 Heinz numbers of integer partitions of odd rank.

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A340931 Heinz numbers of integer partitions of odd numbers into an odd number of parts.

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A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

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A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

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A340786 Number of factorizations of 4n into an even number of even factors > 1.

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A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.