A340692 -id:A340692 - cp's OEIS frontend

A340603 Heinz numbers of integer partitions of odd rank.

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.

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

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

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>PrimeOmega[#]&]

For all terms A061395(a(n)) > A001222(a(n)).

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

A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)      (4*27)        (135)       (4*45)        (4*63)
  (5*9)     (2*6*9)       (3*45)      (12*15)       (12*21)
  (3*15)    (3*4*9)       (5*27)      (4*5*9)       (4*7*9)
  (3*3*5)   (2*2*27)      (9*15)      (2*2*45)      (6*6*7)
            (2*2*3*9)     (3*5*9)     (2*6*15)      (2*2*63)
            (2*2*3*3*3)   (3*3*15)    (3*4*15)      (2*6*21)
                          (3*3*3*5)   (2*2*5*9)     (3*4*21)
                                      (3*3*4*5)     (2*2*7*9)
                                      (2*2*3*15)    (2*3*6*7)
                                      (2*2*3*3*5)   (3*3*4*7)
                                                    (2*2*3*21)
                                                    (2*2*3*3*7)

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

Positions of 0's are A000079.
The version for partitions is A027193.
The version for prime indices is A244991.
The version looking at length instead of greatest factor is A339890.
The version that also has odd length is A340607.
The version looking at least factor is A340832.
- 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.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A058695 counts partitions of odd numbers.
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.

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@*Max]],{n,100}]

A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

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

A340929 Heinz numbers of integer partitions of odd negative rank.

Original entry on oeis.org

4, 12, 16, 18, 27, 40, 48, 60, 64, 72, 90, 100, 108, 112, 135, 150, 160, 162, 168, 192, 225, 240, 243, 250, 252, 256, 280, 288, 352, 360, 375, 378, 392, 400, 420, 432, 448, 528, 540, 567, 588, 600, 625, 630, 640, 648, 672, 700, 768, 792, 810, 832, 880, 882

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

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

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
       4: (1,1)             150: (3,3,2,1)
      12: (2,1,1)           160: (3,1,1,1,1,1)
      16: (1,1,1,1)         162: (2,2,2,2,1)
      18: (2,2,1)           168: (4,2,1,1,1)
      27: (2,2,2)           192: (2,1,1,1,1,1,1)
      40: (3,1,1,1)         225: (3,3,2,2)
      48: (2,1,1,1,1)       240: (3,2,1,1,1,1)
      60: (3,2,1,1)         243: (2,2,2,2,2)
      64: (1,1,1,1,1,1)     250: (3,3,3,1)
      72: (2,2,1,1,1)       252: (4,2,2,1,1)
      90: (3,2,2,1)         256: (1,1,1,1,1,1,1,1)
     100: (3,3,1,1)         280: (4,3,1,1,1)
     108: (2,2,2,1,1)       288: (2,2,1,1,1,1,1)
     112: (4,1,1,1,1)       352: (5,1,1,1,1,1)
     135: (3,2,2,2)         360: (3,2,2,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101707.
The positive version is A101707 (A340604).
The even version is A101708 (A340930).
The not necessarily odd version is A064173 (A340788).
A001222 counts prime factors.
A027193 counts partitions of odd length (A026424).
A047993 counts balanced partitions (A106529).
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank equal to maximum minus minimum part (A324515).
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&]

For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.

A340930 Heinz numbers of integer partitions of even negative rank.

Original entry on oeis.org

8, 24, 32, 36, 54, 80, 81, 96, 120, 128, 144, 180, 200, 216, 224, 270, 300, 320, 324, 336, 384, 405, 450, 480, 486, 500, 504, 512, 560, 576, 675, 704, 720, 729, 750, 756, 784, 800, 840, 864, 896, 1056, 1080, 1125, 1134, 1176, 1200, 1250, 1260, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

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

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
       8: (1,1,1)             270: (3,2,2,2,1)
      24: (2,1,1,1)           300: (3,3,2,1,1)
      32: (1,1,1,1,1)         320: (3,1,1,1,1,1,1)
      36: (2,2,1,1)           324: (2,2,2,2,1,1)
      54: (2,2,2,1)           336: (4,2,1,1,1,1)
      80: (3,1,1,1,1)         384: (2,1,1,1,1,1,1,1)
      81: (2,2,2,2)           405: (3,2,2,2,2)
      96: (2,1,1,1,1,1)       450: (3,3,2,2,1)
     120: (3,2,1,1,1)         480: (3,2,1,1,1,1,1)
     128: (1,1,1,1,1,1,1)     486: (2,2,2,2,2,1)
     144: (2,2,1,1,1,1)       500: (3,3,3,1,1)
     180: (3,2,2,1,1)         504: (4,2,2,1,1,1)
     200: (3,3,1,1,1)         512: (1,1,1,1,1,1,1,1,1)
     216: (2,2,2,1,1,1)       560: (4,3,1,1,1,1)
     224: (4,1,1,1,1,1)       576: (2,2,1,1,1,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101708.
The positive version is (A340605).
The odd version is A101707 (A340929).
The not necessarily even version is A064173 (A340788).
A001222 counts prime factors.
A027187 counts partitions of even length.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058696 counts partitions of even numbers.
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A006141, A039900, A117193, A117409, A316413, A324517, A325134, A326845, A340604, A340787, A340828, A340830.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],EvenQ[rk[#]]&&rk[#]<0&]

A341448 Heinz numbers of integer partitions of type OO.

Original entry on oeis.org

6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240

Offset: 1

Gus Wiseman, Feb 15 2021

nonn

These partitions are defined to have an odd number of odd parts and an odd number of even parts. They also have even length and odd sum.

			The sequence of partitions together with their Heinz numbers begins:
      6: (2,1)         74: (12,1)           141: (15,2)
     14: (4,1)         77: (5,4)            142: (20,1)
     15: (3,2)         86: (14,1)           143: (6,5)
     24: (2,1,1,1)     93: (11,2)           145: (10,3)
     26: (6,1)         95: (8,3)            150: (3,3,2,1)
     33: (5,2)         96: (2,1,1,1,1,1)    152: (8,1,1,1)
     35: (4,3)        104: (6,1,1,1)        158: (22,1)
     38: (8,1)        106: (16,1)           161: (9,4)
     51: (7,2)        119: (7,4)            177: (17,2)
     54: (2,2,2,1)    122: (18,1)           178: (24,1)
     56: (4,1,1,1)    123: (13,2)           185: (12,3)
     58: (10,1)       126: (4,2,2,1)        201: (19,2)
     60: (3,2,1,1)    132: (5,2,1,1)        202: (26,1)
     65: (6,3)        135: (3,2,2,2)        204: (7,2,1,1)
     69: (9,2)        140: (4,3,1,1)        209: (8,5)

Note: A-numbers of ranking sequences are in parentheses below.
The case of odd parts, length, and sum is counted by A078408 (A300272).
The type EE version is A236913 (A340784).
These partitions (for odd n) are counted by A236914.
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd (A340932).
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A160786 counts odd-length partitions of odd numbers (A340931).
A340101 counts factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A000700, A024429, A027187, A106529, A117409, A174725, A257541, A325134, A339890, A340102, A340604.

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

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A340603 Heinz numbers of integer partitions of odd rank.

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A340787 Heinz numbers of integer partitions of positive rank.

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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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A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

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A340929 Heinz numbers of integer partitions of odd negative rank.

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A340930 Heinz numbers of integer partitions of even negative rank.

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A341448 Heinz numbers of integer partitions of type OO.

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