A045931 -id:A045931 - cp's OEIS frontend

A349150 Heinz numbers of integer partitions with at most one odd part.

1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 23, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 77, 78, 79, 81, 83, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109

Offset: 1

Gus Wiseman, Nov 10 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at most one odd prime index.

Also Heinz numbers of partitions with conjugate alternating sum <= 1.

			The terms and their prime indices begin:
      1: {}          23: {9}         49: {4,4}
      2: {1}         26: {1,6}       51: {2,7}
      3: {2}         27: {2,2,2}     53: {16}
      5: {3}         29: {10}        54: {1,2,2,2}
      6: {1,2}       31: {11}        57: {2,8}
      7: {4}         33: {2,5}       58: {1,10}
      9: {2,2}       35: {3,4}       59: {17}
     11: {5}         37: {12}        61: {18}
     13: {6}         38: {1,8}       63: {2,2,4}
     14: {1,4}       39: {2,6}       65: {3,6}
     15: {2,3}       41: {13}        67: {19}
     17: {7}         42: {1,2,4}     69: {2,9}
     18: {1,2,2}     43: {14}        71: {20}
     19: {8}         45: {2,2,3}     73: {21}
     21: {2,4}       47: {15}        74: {1,12}

The case of no odd parts is A066207, counted by A000041 up to 0's.
Requiring all odd parts gives A066208, counted by A000009.
These partitions are counted by A100824, even-length case A349149.
These are the positions of 0's and 1's in A257991.
The conjugate partitions are ranked by A349151.
The case of one odd part is A349158, counted by A000070 up to 0's.
A056239 adds up prime indices, row sums of A112798.
A122111 is a representation of partition conjugation.
A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A340932 ranks partitions whose least part is odd, counted by A026804.
A345958 ranks partitions with alternating sum 1.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.

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

Union of A066207 (no odd parts) and A349158 (one odd part).

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376

Offset: 1

Gus Wiseman, Mar 11 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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 2, 2, 5, 2, 5, 4, 6, 7, 7, 8, 8, 9, 9, 13, 9, 14, 12, 20, 13, 25, 17, 33, 23, 40, 26, 50, 33, 59, 39, 68, 45, 84, 58, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      18         22          28           31              32
   -----------------------------------------------------------------------
    (2,1)  (8,5,3,2)  (8,6,5,3)   (12,7,5,4)   (10,7,5,4,3,2)  (12,8,7,5)
           (8,6,3,1)  (8,7,5,2)   (12,8,5,3)   (10,7,6,5,2,1)  (12,9,7,4)
                      (12,7,2,1)  (12,9,5,2)   (10,8,5,4,3,1)  (16,9,4,3)
                                  (16,9,2,1)   (10,9,6,3,2,1)  (12,10,7,3)
                                  (12,10,5,1)                  (12,11,7,2)
                                                               (16,11,4,1)

The first condition is A239241, non-strict A045931 (ranked by A325698).
This is the strict version of A351977, ranked by A350946.
The second condition is A352129, non-strict A045931 (ranked by A350848).
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, strict A352131.
- A277103, ranked by A350944, strict A000700.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are two other double-pairings of statistics:
- A351976, ranked by A350949.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A014105, A088218, A098123, A195017, A236559, A236914, A241638, A325700, A350839, A350941.

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

A249914 Number of partitions of 4n with equal sums of odd and even parts.

Original entry on oeis.org

1, 1, 4, 12, 30, 70, 165, 330, 704, 1380, 2688, 4984, 9394, 16665, 29970, 52096, 90090, 152064, 257180, 423360, 697851, 1129392, 1819632, 2891520, 4583250, 7162364, 11161752, 17211180, 26427544, 40208520, 60971520, 91641748, 137290956, 204198876, 302530560

Offset: 0

Alois P. Heinz, Feb 11 2015

nonn

			a(0) = 1: [], the empty partition.
a(1) = 1: [2,1,1].
a(2) = 4: [4,3,1], [4,1,1,1,1], [3,2,2,1], [2,2,1,1,1,1].
a(3) = 12: [6,5,1], [6,3,3], [6,3,1,1,1], [6,1,1,1,1,1,1], [5,4,2,1], [5,2,2,2,1], [4,3,3,2], [4,3,2,1,1,1], [4,2,1,1,1,1,1,1], [3,3,2,2,2], [3,2,2,2,1,1,1], [2,2,2,1,1,1,1,1,1].

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

Cf. A000009, A000041, A035294, A045931, A255001.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> combinat[numbpart](n) *b(2*n, 2*n-1):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]]];
a[n_] := PartitionsP[n] b[2n, 2n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = A000041(n) * A035294(n) = A000041(n) * A000009(2n).

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*6^(3/4)*n^(7/4)). - Vaclav Kotesovec, Dec 11 2020

A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

Original entry on oeis.org

1, 10, 39, 88, 115, 228, 259, 306, 517, 544, 620, 783, 793, 870, 1150, 1204, 1241, 1392, 1656, 1691, 1722, 1845, 2369, 2590, 2596, 2775, 2944, 3038, 3277, 3280, 3339, 3498, 3692, 3996, 4247, 4440, 4935, 5022, 5170, 5226, 5587, 5644, 5875, 5936, 6200, 6321

Offset: 1

Gus Wiseman, Nov 26 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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their negated alternating sum.

			The terms and their prime indices begin:
     1: ()
    10: (3,1)
    39: (6,2)
    88: (5,1,1,1)
   115: (9,3)
   228: (8,2,1,1)
   259: (12,4)
   306: (7,2,2,1)
   517: (15,5)
   544: (7,1,1,1,1,1)
   620: (11,3,1,1)
   783: (10,2,2,2)
   793: (18,6)
   870: (10,3,2,1)
  1150: (9,3,3,1)
  1204: (14,4,1,1)
  1241: (21,7)
  1392: (10,2,1,1,1,1)
  1656: (9,2,2,1,1,1)
  1691: (24,8)

These partitions are counted by A001523 up to 0's.
An ordered version is A349154, nonnegative A348614, reverse A349155.
The nonnegative version is A349159, counted by A000712 up to 0's.
The reverse nonnegative version is A349160, counted by A006330 up to 0's.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, A345197 count compositions by alternating sum.
A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000984, A001700, A028260, A045931, A120452, A195017, A257991, A257992, A262977, A325698, A344619, A345958.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==-2*ats[primeMS[#]]&]

A056239(a(n)) = -2*A316524(a(n)).

A346698(a(n)) = 3*A346697(a(n)).

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Gus Wiseman, Mar 18 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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Intersection of A066208 and A346635.

A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

Original entry on oeis.org

1, 12, 70, 90, 112, 144, 286, 325, 462, 520, 525, 594, 646, 675, 832, 840, 1045, 1080, 1326, 1334, 1344, 1666, 1672, 1728, 1900, 2142, 2145, 2294, 2465, 2622, 2695, 2754, 3040, 3432, 3465, 3509, 3526, 3900, 3944, 4186, 4255, 4312, 4455, 4845, 4864, 4900, 4982

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 terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    70: {1,3,4}
    90: {1,2,2,3}
   112: {1,1,1,1,4}
   144: {1,1,1,1,2,2}
   286: {1,5,6}
   325: {3,3,6}
   462: {1,2,4,5}
   520: {1,1,1,3,6}
   525: {2,3,3,4}
   594: {1,2,2,2,5}
   646: {1,7,8}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
   840: {1,1,1,2,3,4}
For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.

For prime factors instead of indices we have A019507.
Partitions of this type are counted by A239261.
For count instead of sum we have A325698, distinct A325700.
The LHS (sum of odd prime indices) is A366528, triangle A113685.
The RHS (sum of even prime indices) is A366531, triangle A113686.
These are the positions of zeros in A366749.
A000009 counts partitions into odd parts, ranked by A066208.
A035363 counts partitions into even parts, ranked by A066207.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A106529, A239241, A241638, A365067, A366533.

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

These are numbers k such that A346697(k) = A346698(k).

A349149 Number of even-length integer partitions of n with at most one odd part in the conjugate partition.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 7, 12, 11, 19, 15, 30, 22, 45, 30, 67, 42, 97, 56, 139, 77, 195, 101, 272, 135, 373, 176, 508, 231, 684, 297, 915, 385, 1212, 490, 1597, 627, 2087, 792, 2714, 1002, 3506, 1255, 4508, 1575, 5763, 1958, 7338, 2436, 9296, 3010, 11732

Offset: 0

Gus Wiseman, Nov 09 2021

nonn

The alternating sum of a partition is equal to the number of odd parts in the conjugate partition, so this sequence counts even-length partitions with alternating sum <= 1.

			The a(2) = 1 through a(9) = 7 partitions:
  11   21   22     32     33       43       44         54
            1111   2111   2211     2221     2222       3222
                          111111   3211     3311       3321
                                   211111   221111     4311
                                            11111111   222111
                                                       321111
                                                       21111111

The case of 0 odd conjugate parts is A000041 up to 0's, ranked by A000290.
The case of 1 odd conjugate part is A000070 up to 0's.
Even bisection of A100824, ranked by A349150.
Ranked by A349151 /\ A028260.
A045931 counts partitions with as many even as odd parts, ranked by A325698.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A122111 is a representation of partition conjugation.
A277103 counts partitions with the same alternating sum as their conjugate.
A277579 counts partitions with as many even parts as odd conjugate parts.
A325039 counts partitions with the same product as their conjugate.
A344610 counts partitions by sum and positive reverse-alternating sum.
A345196 counts partitions with the same rev-alt sum as their conjugate.
Cf. A000097, A000700, A001700, A027187, A027193, A108711, A236559, A236913, A325534, A344607, A344651.

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

a(2n) = A000041(n).

a(2n+1) = A000070(n-1).

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

cp's OEIS Frontend

A349150 Heinz numbers of integer partitions with at most one odd part.

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A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

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A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

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A249914 Number of partitions of 4n with equal sums of odd and even parts.

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A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

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A352143 Numbers whose prime indices and conjugate prime indices are all odd.

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A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

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A349149 Number of even-length integer partitions of n with at most one odd part in the conjugate partition.

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