A058695 -id:A058695 - cp's OEIS frontend

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

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

A100824 Number of partitions of n with at most one odd part.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742

Offset: 0

Vladeta Jovovic, Jan 13 2005

From Gus Wiseman, Jan 21 2022: (Start)
Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
  1  11  21   22    32     33      43       44        54
         111  1111  221    2211    331      2222      441
                    2111   111111  2221     3311      3222
                    11111          3211     221111    3321
                                   22111    11111111  4311
                                   211111             22221
                                   1111111            33111
                                                      222111
                                                      321111
                                                      2211111
                                                      21111111
                                                      111111111
(End)

			From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)     (9)
            (21)  (22)  (32)   (42)   (43)    (44)    (54)
                        (41)   (222)  (52)    (62)    (63)
                        (221)         (61)    (422)   (72)
                                      (322)   (2222)  (81)
                                      (421)           (432)
                                      (2221)          (441)
                                                      (522)
                                                      (621)
                                                      (3222)
                                                      (4221)
                                                      (22221)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A008951, A000070, A000097, A000098, A000710.
The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
These partitions are ranked by A349150, conjugate A349151.
A000041 = integer partitions, strict A000009.
A027187 = partitions of even length, strict A067661, ranked by A028260.
A027193 = partitions of odd length, ranked by A026424.
A058695 = partitions of odd numbers.
A103919 = partitions by sum and alternating sum (reverse: A344612).
A277103 = partitions with the same number of odd parts as their conjugate.
Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.

seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)

a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

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

A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 19, 25, 27, 31, 33, 37, 41, 45, 51, 55, 57, 59, 61, 67, 71, 75, 81, 83, 85, 93, 95, 99, 103, 107, 109, 111, 113, 121, 123, 125, 127, 131, 135, 153, 155, 157, 165, 171, 177, 179, 181, 183, 185, 187, 191, 193, 197, 201, 205, 209, 211, 213

Offset: 1

Gus Wiseman, Sep 12 2022

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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multiset partitions:
   1: {}
   3: {{1}}
   5: {{2}}
   9: {{1},{1}}
  11: {{3}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  45: {{1},{1},{2}}
  51: {{1},{4}}
  55: {{2},{3}}
  57: {{1},{1,1,1}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Odd-size multisets are ctd by A000302, A027193, A058695, rkd by A026424.
Other types: A050330, A356932, A356933, A356934.
Other conditions: A302478, A302492, A356930, A356939, A356940, A356944, A356945.
Cf. A000040, A000720, A003963, A270995, A302242, A304969, A349050, A349055.

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

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

Original entry on oeis.org

2, 6, 7, 8, 9, 10, 11, 13, 15, 19, 24, 28, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 54, 57, 58, 59, 60, 61, 65, 67, 70, 73, 76, 77, 79, 85, 86, 90, 95, 96, 97, 98, 103, 106, 107, 109, 110, 111, 112, 117, 119, 123, 124, 126, 127, 128, 129

Offset: 1

Gus Wiseman, May 14 2024

nonn

The odd version is A372590.

			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)
        {1,4}   9  (2,2)
        {2,4}  10  (3,1)
      {1,2,4}  11  (5)
      {1,3,4}  13  (6)
    {1,2,3,4}  15  (3,2)
      {1,2,5}  19  (8)
        {4,5}  24  (2,1,1,1)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
          {6}  32  (1,1,1,1,1)
        {1,6}  33  (5,2)
        {2,6}  34  (7,1)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
    {1,2,3,6}  39  (6,2)
        {4,6}  40  (3,1,1,1)
      {1,4,6}  41  (13)
      {2,4,6}  42  (4,2,1)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
Positions of even terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372590.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
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, A006141, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

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

A154795 Odd partition numbers of odd numbers.

Original entry on oeis.org

1, 3, 7, 15, 101, 297, 1255, 4565, 10143, 14883, 21637, 31185, 44583, 63261, 173525, 239943, 329931, 1121505, 1505499, 2679689, 3554345, 4697205, 6185689, 10619863, 18004327, 23338469, 30167357, 38887673, 49995925, 64112359, 82010177

Offset: 1

Omar E. Pol, Jan 26 2009

nonn

Odd numbers in A058695.

			7 is in the sequence because the odd number 5 has partition number 7 (5,41,32,311,2221,22111,1111111). - _Emeric Deutsch_, Aug 02 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A005408, A058695, A154796, A154797, A154798.

aa:= proc(n, i) if n=0 then 1 elif n<0 or i=0 then 0 else aa(n,i):= aa(n, i-1) +aa(n-i, i) fi end: a:= proc(n) local k; if n>1 then a(n-1) fi; for k from `if`(n=1, 1, b(n-1)+2) by 2 while irem(aa(k, k), 2)=0 do od; b(n):= k; aa(k, k) end: seq(a(n), n=1..40); # Alois P. Heinz, Jul 28 2009
with(combinat): a := proc (n) if `mod`(numbpart(2*n-1), 2) = 1 then numbpart(2*n-1) else end if end proc: seq(a(n), n = 1 .. 50); # Emeric Deutsch, Aug 02 2009

Reap[Do[If[OddQ[p = PartitionsP[n]], Sow[p]], {n, 1, 99, 2}]][[2, 1]] (* Jean-François Alcover, Aug 31 2015 *)

More terms from Alois P. Heinz, Jul 28 2009

A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

Original entry on oeis.org

3, 7, 9, 13, 15, 19, 21, 27, 29, 33, 37, 39, 43, 45, 49, 51, 53, 57, 61, 63, 69, 71, 75, 77, 79, 81, 87, 89, 91, 93, 99, 101, 105, 107, 111, 113, 117, 119, 123, 129, 131, 133, 135, 139, 141, 147, 151, 153, 159, 161, 163, 165, 169, 171, 173, 177, 181, 183

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 counted.

			The sequence of terms together with their prime indices begins:
      3: {2}         51: {2,7}         99: {2,2,5}
      7: {4}         53: {16}         101: {26}
      9: {2,2}       57: {2,8}        105: {2,3,4}
     13: {6}         61: {18}         107: {28}
     15: {2,3}       63: {2,2,4}      111: {2,12}
     19: {8}         69: {2,9}        113: {30}
     21: {2,4}       71: {20}         117: {2,2,6}
     27: {2,2,2}     75: {2,3,3}      119: {4,7}
     29: {10}        77: {4,5}        123: {2,13}
     33: {2,5}       79: {22}         129: {2,14}
     37: {12}        81: {2,2,2,2}    131: {32}
     39: {2,6}       87: {2,10}       133: {4,8}
     43: {14}        89: {24}         135: {2,2,2,3}
     45: {2,2,3}     91: {4,6}        139: {34}
     49: {4,4}       93: {2,11}       141: {2,15}

These partitions are counted by A026805.
Looking at length or at maximum gives A028260/A244990, counted by A027187.
If all prime indices are even we get A066207, counted by A035363.
The complement is {1} \/ A340932, counted by A026804.
A001222 counts prime factors.
A005843 lists even numbers.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058695 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A006141, A030229, A067661, A101708, A236913,  A339846, A340601, A340602, A340605, A340854/A340855.

Select[Range[2,100],EvenQ[PrimePi[FactorInteger[#][[1,1]]]]&]

A055396(a(n)) belongs to A005843.

Closed under multiplication.

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.

A154796 Even partition numbers of odd numbers.

Original entry on oeis.org

30, 56, 176, 490, 792, 1958, 3010, 6842, 89134, 124754, 451276, 614154, 831820, 2012558, 8118264, 13848650, 133230930, 214481126, 271248950, 541946240, 851376628, 1327710076, 3163127352, 4835271870, 5964539504, 7346629512

Offset: 1

Omar E. Pol, Jan 26 2009

nonn

Even numbers in A058695.

			The even number 30 is in the sequence as the partition number of the odd number 9. - _Emeric Deutsch_, Aug 02 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A005408, A058695, A154795, A154797, A154798.

aa:= proc(n, i) if n=0 then 1 elif n<0 or i=0 then 0 else aa(n,i):= aa(n, i-1) +aa(n-i, i) fi end: a:= proc(n) local k; if n>1 then a(n-1) fi; for k from `if`(n=1, 1, b(n-1)+2) by 2 while irem(aa(k, k), 2)=1 do od; b(n):= k; aa(k, k) end: seq(a(n), n=1..40); # Alois P. Heinz, Jul 28 2009
with(combinat): a := proc (n) if `mod`(numbpart(2*n-1), 2) = 0 then numbpart(2*n-1) else end if end proc: seq(a(n), n = 1 .. 70); # Emeric Deutsch, Aug 02 2009

Reap[Do[If[EvenQ[p = PartitionsP[n]], Sow[p]], {n, 1, 199, 2}]][[2, 1]] (* Jean-François Alcover, Nov 11 2015 *)
Select[PartitionsP[Range[1,201,2]],EvenQ] (* Harvey P. Dale, Apr 03 2019 *)

lista(nn) = for (n=1, nn, if (((p = numbpart(2*n+1)) % 2) == 0, print1(p, ", "))); \\ Michel Marcus, Dec 19 2016

More terms from Alois P. Heinz, Jul 28 2009

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.

cp's OEIS Frontend

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

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A100824 Number of partitions of n with at most one odd part.

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

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A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

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A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

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A154795 Odd partition numbers of odd numbers.

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A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

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