A347450 -id:A347450 - cp's OEIS frontend

A088902 Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 350, 416, 441, 624, 660, 735, 1088, 1100, 1386, 1560, 1632, 1715, 2310, 2401, 2432, 2600, 3267, 3276, 3648, 4080, 5390, 5445, 5460, 5888, 6800, 7546, 7722, 8568, 8832, 9120, 12705, 12740, 12870, 13689

Offset: 1

Naohiro Nomoto, Nov 28 2003

The Heinz numbers of the self-conjugate partitions. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] to be Product(p_j-th prime, j=1..r) (a concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56. It is in the sequence since [1,1,1,4] is self-conjugate. - Emeric Deutsch, Jun 05 2015

			20 is in the sequence because 20 = 2^2 * 5^1 = (p_1)^2 *(p_3)^1, (two 1's, one 3's) = (1,1,3) is a self-conjugate partition of 5.
From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their prime indices begin:
    1: ()
    2: (1)
    6: (2,1)
    9: (2,2)
   20: (3,1,1)
   30: (3,2,1)
   56: (4,1,1,1)
   75: (3,3,2)
   84: (4,2,1,1)
  125: (3,3,3)
  176: (5,1,1,1,1)
  210: (4,3,2,1)
  264: (5,2,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..400

Fixed points of A122111.
Cf. A242422, A215366.
A002110 (primorial numbers) is a subsequence.
After a(1) and a(2), a subsequence of A241913.
These partitions are counted by A000700.
The same count comes from A258116.
The complement is A352486, counted by A330644.
These are the positions of zeros in A352491.
A000041 counts integer partitions, strict A000009.
A325039 counts partitions w/ product = conjugate product, ranked by A325040.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382.
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition.
- A296150 = parts of partition, reverse A112798, conjugate A321649.
- A352487 = less than conjugate, counted by A000701.
- A352488 = greater than or equal to conjugate, counted by A046682.
- A352489 = less than or equal to conjugate, counted by A046682.
- A352490 = greater than conjugate, counted by A000701.
Cf. A000720, A098825, A195017, A238351, A238745, A347450, A350841.

with(numtheory): c := proc (n) local B, C: B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: C := proc (P) local a: a := proc (j) local c, i: c := 0: for i to nops(P) do if j <= P[i] then c := c+1 else end if end do: c end proc: [seq(a(k), k = 1 .. max(P))] end proc: mul(ithprime(C(B(n))[q]), q = 1 .. nops(C(B(n)))) end proc: SC := {}: for i to 14000 do if c(i) = i then SC := `union`(SC, {i}) else end if end do: SC; # Emeric Deutsch, May 09 2015

Select[Range[14000], Function[n, n == If[n == 1, 1, Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger@ n]]]] (* Michael De Vlieger, Aug 27 2016, after JungHwan Min at A122111 *)

More terms from David Wasserman, Aug 26 2005

A347446 Number of integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 54, 62, 84, 100, 134, 157, 207, 241, 314, 363, 463, 537, 685, 785, 985, 1138, 1410, 1616, 1996, 2286, 2801, 3201, 3885, 4434, 5363, 6098, 7323, 8329, 9954, 11293, 13430, 15214, 18022, 20383, 24017, 27141, 31893, 35960

Offset: 0

Gus Wiseman, Sep 15 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (322)
                    (211)   (311)    (51)      (331)
                    (1111)  (2111)   (222)     (421)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

Allowing any reverse-alternating product >= 1 gives A344607.
Allowing any alternating product <= 1 gives A119620, reverse A347443.
Allowing any reverse-alternating product < 1 gives A344608.
The multiplicative version (factorizations) is A347437, reverse A347442.
The odd-length case is A347444, ranked by A347453.
The reverse version is A347445, ranked by A347454.
Allowing any alternating product > 1 gives A347448, reverse A347449.
Ranked by A347457.
The even-length case is A347704.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A347461 counts possible alternating products of partitions.
Cf. A025047, A067661, A339890, A347450, A344654, A344740, A347439, A347440, A347451, A347463, A347705.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],IntegerQ[altprod[#]]&]],{n,0,30}]

A347456 Number of factorizations of n with alternating product >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 6, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 8, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 4, 1, 1, 5, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 09 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Also the number of factorizations of n with alternating sum >= 0.

			The a(n) factorizations for n = 4, 16, 24, 36, 60, 64, 96:
  4     16        24      36        60       64            96
  2*2   4*4       2*2*6   6*6       2*5*6    8*8           2*6*8
        2*2*4     2*3*4   2*2*9     3*4*5    2*4*8         3*4*8
        2*2*2*2           2*3*6     2*2*15   4*4*4         4*4*6
                          3*3*4     2*3*10   2*2*16        2*2*24
                          2*2*3*3            2*2*4*4       2*3*16
                                             2*2*2*2*4     2*4*12
                                             2*2*2*2*2*2   2*2*2*2*6
                                                           2*2*2*3*4

The case of partitions is A000041, reverse A344607.
The reverse version is A001055, strict A347705.
Positions of 3's appear to be A065036.
Positions of 1's are 1 and A167171.
The opposite version (<= instead of >=) is A339846.
The strict version (> instead of >=) is A339890, also the odd-length case.
Allowing any integer alternating product gives A347437.
The case of alternating product 1 is A347438, also the even-length case.
Allowing any integer reciprocal alternating product gives A347439.
The complement (< instead of >=) is A347440.
Allowing any integer reverse-alternating product gives A347442.
A038548 counts factorizations with a wiggly permutation.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A347447 counts strict factorizations with alternating product > 1.
Cf. A001700, A028983, A316523, A347441, A347443, A347446, A347448, A347450, A347454, A347463, A347708.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],altprod[#]>=1&]],{n,100}]

a(n) = A347438(n) + A347440(n).

A347457 Heinz numbers of integer partitions with integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 75, 76, 78

Offset: 1

Gus Wiseman, Sep 26 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has integer reverse-alternating product.

			The prime indices of 525 are {2,3,3,4}, with reverse-alternating product 2, so 525 is in the sequence
The prime indices of 135 are {2,2,2,3}, with reverse-alternating product 3/2, so 135 is not in the sequence.

The reciprocal version is A028982.
Allowing any alternating product > 1 gives A028983, reverse A347465.
Factorizations of this type are counted by A347437.
These partitions are counted by A347446.
The reverse reciprocal version A347451.
The odd-length case is A347453.
The reverse version is A347454.
The complement is A347455.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A347461 counts possible alternating products of partitions, reverse A347462.
Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[altprod[Reverse[primeMS[#]]]]&]

A347445 Number of integer partitions of n with integer reverse-alternating product.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 24, 32, 40, 50, 62, 77, 99, 115, 151, 170, 224, 251, 331, 360, 481, 517, 690, 728, 980, 1020, 1379, 1420, 1918, 1962, 2643, 2677, 3630, 3651, 4920, 4926, 6659, 6625, 8931, 8853, 11905, 11781, 15805, 15562, 20872, 20518

Offset: 0

Gus Wiseman, Sep 14 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (411)     (421)      (422)
                                     (2211)    (511)      (611)
                                     (21111)   (22111)    (2222)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (22211)
                                                          (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Allowing any reverse-alternating product >= 1 gives A344607.
Allowing any reverse-alternating product < 1 gives A344608.
The multiplicative version is A347442, unreversed A347437.
Allowing any reverse-alternating product <= 1 gives A347443.
Restricting to odd length gives A347444, ranked by A347453.
The unreversed version is A347446, ranked by A347457.
Allowing any reverse-alternating product > 1 gives A347449.
Ranked by A347454.
A000041 counts partitions, with multiplicative version A001055.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A339890 counts factorizations with alternating product > 1, reverse A347705.
A347462 counts possible reverse-alternating products of partitions.
Cf. A025047, A067661, A119620, A344654, A344740, A347439, A347440, A347448, A347450, A347451, A347461, A347463, A347704.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],IntegerQ[revaltprod[#]]&]],{n,0,30}]

A347443 Number of integer partitions of n with reverse-alternating product <= 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 12, 19, 22, 34, 40, 60, 69, 101, 118, 168, 195, 272, 317, 434, 505, 679, 793, 1050, 1224, 1599, 1867, 2409, 2811, 3587, 4186, 5290, 6168, 7724, 9005, 11186, 13026, 16062, 18692, 22894, 26613, 32394, 37619, 45535, 52815, 63593, 73680

Offset: 0

Gus Wiseman, Sep 14 2021

nonn

Includes all partitions of even length (A027187).
Also the number of integer partitions of n with reverse-alternating sum <= 1.
Also the number of integer partitions of n having either even length (A027187) or having exactly one odd part in the conjugate partition (A100824).
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (2111)   (2211)    (331)      (71)
                            (11111)  (3111)    (2221)     (2222)
                                     (111111)  (3211)     (3221)
                                               (4111)     (3311)
                                               (22111)    (4211)
                                               (211111)   (5111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (11111111)

The odd-length case is A035363 (shifted).
The strict case is A067661.
The non-reverse version is counted by A119620, ranked by A347466.
The even bisection is A236913.
The opposite version (>= instead of <=) is A344607.
The case of < 1 instead of <= 1 is A344608.
The multiplicative version (factorizations) is A347438, non-reverse A339846.
Allowing any integer reverse-alternating product gives A347445.
The complement (> 1 instead of <= 1) is counted by A347449.
Ranked by A347465, non-reverse A347450.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A058622 counts compositions with alternating sum <= 0 (A294175 for < 0).
A100824 counts partitions with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A038548, A086543, A116406, A325534, A325535, A344611, A344654, A344740, A347440, A347442, A347446, A347448.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]<=1&]],{n,0,30}]

a(n) = A027187(n) + A035363(n-1) for n >= 1. [Corrected by Georg Fischer, Dec 13 2022]

a(n) = A119620(n) + A344608(n).

A347454 Numbers whose multiset of prime indices has integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113

Offset: 1

Gus Wiseman, Sep 26 2021

nonn

First differs from A265640 in having 42.
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 alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers of partitions with integer reverse-alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The terms and their prime indices begin:
      1: {}            20: {1,1,3}         47: {15}
      2: {1}           23: {9}             48: {1,1,1,1,2}
      3: {2}           25: {3,3}           49: {4,4}
      4: {1,1}         27: {2,2,2}         50: {1,3,3}
      5: {3}           28: {1,1,4}         52: {1,1,6}
      7: {4}           29: {10}            53: {16}
      8: {1,1,1}       31: {11}            59: {17}
      9: {2,2}         32: {1,1,1,1,1}     61: {18}
     11: {5}           36: {1,1,2,2}       63: {2,2,4}
     12: {1,1,2}       37: {12}            64: {1,1,1,1,1,1}
     13: {6}           41: {13}            67: {19}
     16: {1,1,1,1}     42: {1,2,4}         68: {1,1,7}
     17: {7}           43: {14}            71: {20}
     18: {1,2,2}       44: {1,1,5}         72: {1,1,1,2,2}
     19: {8}           45: {2,2,3}         73: {21}

The even-length case is A000290.
The additive version is A026424.
Allowing any alternating product < 1 gives A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Factorizations of this type are counted by A347437.
These partitions are counted by A347445, reverse A347446.
Allowing any alternating product <= 1 gives A347450.
The reciprocal version is A347451.
The odd-length case is A347453.
The version for reversed prime indices is A347457, complement A347455.
Allowing any alternating product > 1 gives A347465, reverse A028983.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[altprod[primeMS[#]]]&]

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, -1, 1, -3, 0, -9, 3, 0, -2, -21, 2, -51, -10, -3, 9, -111, 3, -237, 0, -15, -26, -489, 10, -2, -70, 2, -12, -995, 0, -2017, 21, -39, -158, -19, 15, -4059, -346, -105, 12, -8151, -18, -16341, -36, -5, -722, -32721, 26, -32, 5, -237, -108, -65483, 19, -53

Offset: 1

Gus Wiseman, Mar 20 2022

sign

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.

Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...

			The partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, so a(196) = 196 - 189 = 7.

Positions of zeros are A088902, counted by A000700.
A similar sequence is A175508.
Positions of nonzero terms are A352486, counted by A330644.
Positions of negative terms are A352487, counted by A000701.
Positions of nonnegative terms are A352488, counted by A046682.
Positions of nonpositive terms are A352489, counted by A046682.
Positions of positive terms are A352490, counted by A000701.
A000041 counts integer partitions, strict A000009.
A003963 is product of prime indices, conjugate A329382.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A238744 is partition conjugate of prime signature, ranked by A238745.
Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, A347450.

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]}]];
Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

a(n) = n - A122111(n).

A350946 Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 6, 65, 84, 210, 216, 319, 490, 525, 532, 731, 1254, 1403, 1924, 2184, 2340, 2449, 2470, 3024, 3135, 3325, 3774, 4028, 4141, 4522, 5311, 5460, 7030, 7314, 7315, 7560, 7776, 7942, 8201, 8236, 9048, 9435, 9464, 10659, 10921, 11484, 11914, 12012, 12025, 12740

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

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

			The terms together with their prime indices begin:
     1: ()
     6: (2,1)
    65: (6,3)
    84: (4,2,1,1)
   210: (4,3,2,1)
   216: (2,2,2,1,1,1)
   319: (10,5)
   490: (4,4,3,1)
   525: (4,3,3,2)
   532: (8,4,1,1)
   731: (14,7)
  1254: (8,5,2,1)
  1403: (18,9)
  1924: (12,6,1,1)
  2184: (6,4,2,1,1,1)
  2340: (6,3,2,2,1,1)
  2449: (22,11)
  2470: (8,6,3,1)
For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to the sequence.

For the first condition alone:
- counted by A045931 (strict A239241)
- ordered version (compositions) A098123
- ranked by A325698
- without multiplicity A325700 (counted by A241638)
The second condition alone is ranked by A350848, strict A352129.
These partitions are counted by A351977, strict A352128.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350943: # of even conj parts = # of odd parts, strict counted by A352130.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are two other possible double-pairings of statistics:
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A130780, A171966, A347450, A350849, A350941, A350942, A350950, A350951.

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[1000],#==1||Mean[Mod[primeMS[#],2]]== Mean[Mod[conj[primeMS[#]],2]]==1/2&]

Closed under A122111 (conjugation).
Intersection of A325698 and A350848.
A257992(a(n)) = A257991(a(n)).
A350847(a(n)) = A344616(a(n)).

A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, 0, -3, 0, 3, 1, 1, 0, 0, 0, -1, -1, -2, 0, 1, 0, 0, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, -2, -1, 1, 1, 2, -3, 0, 0, 0, 0, -1, 1, -1, 3, -1, -2, 0, 0, 0, -1, -1, 1, 1, 0, 0, 0, 1, -3, 1, 1, 0

Offset: 1

Gus Wiseman, Mar 14 2022

sign

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 prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 2 - 1 = 1.

The version comparing even with odd parts is A195017.
The version comparing even with odd conjugate parts is A350849.
The version comparing even conjugate with odd conjugate parts is A350941.
The version comparing odd with even conjugate parts is A350942.
Positions of 0's are A350945, counted by A350948.
The version comparing odd with odd conjugate parts is A350951.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are five other possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Cf. A028260, A088218, A098123, A236559, A239241, A241638, A325700, A347450.

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]}]];
Table[Count[primeMS[n],?EvenQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

a(n) = A257992(n) - A350847(n).

a(A122111(n)) = -a(n), where A122111 represents partition conjugation.

cp's OEIS Frontend

A088902 Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.

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A347446 Number of integer partitions of n with integer alternating product.

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A347456 Number of factorizations of n with alternating product >= 1.

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A347457 Heinz numbers of integer partitions with integer alternating product.

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A347445 Number of integer partitions of n with integer reverse-alternating product.

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A347443 Number of integer partitions of n with reverse-alternating product <= 1.

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A347454 Numbers whose multiset of prime indices has integer alternating product.

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A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

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A350946 Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.