A325040 -id:A325040 - 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

A325036 Difference between product and sum of prime indices of n.

Original entry on oeis.org

1, 0, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -1, 1, -3, 0, -1, 0, -2, 2, -1, 0, -3, 3, -1, 2, -2, 0, 0, 0, -4, 3, -1, 5, -2, 0, -1, 4, -3, 0, 1, 0, -2, 5, -1, 0, -4, 8, 2, 5, -2, 0, 1, 7, -3, 6, -1, 0, -1, 0, -1, 8, -5, 9, 2, 0, -2, 7, 4, 0, -3, 0, -1, 10, -2, 11, 3, 0, -4, 8, -1, 0, 0, 11, -1, 8, -3, 0, 4, 14, -2, 9

Offset: 1

Gus Wiseman, Mar 25 2019

sign

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 prime indices of 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.
Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.
Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@primeMS[n]-Total[primeMS[n]],{n,100}]
dps[n_]:=Module[{pi=Flatten[Table[PrimePi[#[[1]]],#[[2]]]&/@FactorInteger[n]]},Times@@pi-Total[pi]]; Join[{1},Array[dps,100,2]] (* Harvey P. Dale, May 26 2023 *)

A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };
A325036(n) = (A003963(n) - A056239(n)); \\ Antti Karttunen, May 08 2022

a(n) = A003963(n) - A056239(n).

For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - Antti Karttunen, May 08 2022

Data section extended up to a(93) by Antti Karttunen, May 08 2022

A352486 Heinz numbers of non-self-conjugate integer partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Mar 20 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 sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.

			The terms together with their prime indices begin:
   3: (2)
   4: (1,1)
   5: (3)
   7: (4)
   8: (1,1,1)
  10: (3,1)
  11: (5)
  12: (2,1,1)
  13: (6)
  14: (4,1)
  15: (3,2)
  16: (1,1,1,1)
  17: (7)
  18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.

The complement is A088902, counted by A000700.
These partitions are counted by A330644.
These are the positions of nonzero terms in A352491.
A000041 counts integer partitions, strict A000009.
A098825 counts permutations by unfixed points.
A238349 counts compositions by fixed points, rank statistic A352512.
A325039 counts partitions w/ same product as conjugate, ranked by A325040.
A352523 counts compositions by unfixed points, rank statistic A352513.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition
- A122111 = rank of conjugate 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, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524,  A324846, A350841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]

a(n) != A122111(a(n)).

A350945 Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 36, 38, 39, 41, 44, 47, 56, 57, 58, 59, 66, 67, 68, 73, 74, 75, 80, 83, 84, 86, 87, 92, 96, 97, 102, 103, 104, 106, 109, 111, 120, 122, 124, 125, 127, 128, 129, 137, 138, 142, 144, 149, 152, 156

Offset: 1

Gus Wiseman, Jan 28 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: ()
   2: (1)
   5: (3)
   6: (2,1)
   8: (1,1,1)
   9: (2,2)
  11: (5)
  14: (4,1)
  17: (7)
  20: (3,1,1)
  21: (4,2)
  23: (9)
  24: (2,1,1,1)

These partitions are counted by A350948.
These are the positions of 0's in A350950.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  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.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, `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[100],Count[conj[primeMS[#]],?EvenQ]==Count[primeMS[#],?EvenQ]&]

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

A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 15, 18, 20, 30, 35, 49, 54, 55, 56, 70, 75, 77, 81, 84, 88, 90, 98, 108, 110, 112, 125, 132, 135, 143, 154, 162, 168, 169, 176, 180, 187, 210, 221, 260, 264, 270, 286, 294, 315, 323, 330, 338, 340, 350, 361, 363, 364, 374, 391, 416, 420

Offset: 1

Gus Wiseman, Jan 28 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: ()
   2: (1)
   6: (2,1)
   9: (2,2)
  10: (3,1)
  12: (2,1,1)
  15: (3,2)
  18: (2,2,1)
  20: (3,1,1)
  30: (3,2,1)
  35: (4,3)
  49: (4,4)
  54: (2,2,2,1)

These partitions are counted by A277103.
The even rank case is A345196.
The conjugate version is A350943, counted by A277579.
These are the positions of 0's in A350951, even A350950.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  A325698: # of even parts = # of odd parts, counted by A045931.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # odd conj parts, counted by A045931.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000700, A027187, A027193, A103919, A236559, A350942.

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],Count[conj[primeMS[#]],?OddQ]==Count[primeMS[#],?OddQ]&]

A257991(a(n)) = A344616(a(n)).

A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

Original entry on oeis.org

1, 3, 6, 7, 13, 14, 18, 19, 26, 27, 29, 36, 37, 38, 42, 43, 53, 54, 58, 61, 63, 70, 71, 74, 78, 79, 84, 86, 89, 101, 105, 106, 107, 113, 114, 117, 122, 126, 130, 131, 139, 140, 142, 151, 156, 158, 162, 163, 171, 173, 174, 178, 181, 190, 193, 195, 199, 202, 210

Offset: 1

Gus Wiseman, Jan 28 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: ()
   3: (2)
   6: (2,1)
   7: (4)
  13: (6)
  14: (4,1)
  18: (2,2,1)
  19: (8)
  26: (6,1)
  27: (2,2,2)
  29: (10)
  36: (2,2,1,1)
  37: (12)
  38: (8,1)
  42: (4,2,1)
For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 is in the sequence.

These partitions are counted by A277579.
The conjugate version is A349157, also counted by A277579.
These are the positions of 0's in A350942.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  A325698: # of even parts = # of odd parts, counted by A045931.
  A350848: # of even conj parts = # of odd conj parts, counted by A045931.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, 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[100],Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]&]

A350847(a(n)) = A257991(a(n)).

A325039 Number of integer partitions of n with the same product of parts as their conjugate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 6, 2, 2, 4, 3, 5, 7, 6, 5, 7, 9, 10, 11, 18, 16, 19, 19, 16, 20, 20, 28, 39, 28, 40, 53, 45, 52, 59, 71, 61, 73, 97, 102, 95, 112, 131, 137, 148, 140, 166, 199, 181, 238, 251, 255, 289, 339, 344, 381, 398, 422, 464, 541, 555, 628, 677, 732

Offset: 0

Gus Wiseman, Mar 25 2019

nonn

For example, the partition (6,4,1) with product 24 has conjugate (3,2,2,2,1,1) with product also 24.

The Heinz numbers of these partitions are given by A325040.

			The a(8) = 6 through a(15) = 6 integer partitions:
  (44)    (333)    (4321)   (641)     (4422)    (4432)     (6431)
  (332)   (51111)  (52111)  (4331)    (53211)   (6421)     (8411)
  (431)                     (322211)  (621111)  (53311)    (54221)
  (2222)                    (611111)            (432211)   (433211)
  (3221)                                        (7111111)  (632111)
  (4211)                                                   (7211111)
                                                           (42221111)

Cf. A001055, A064573, A122111, A296150, A318950, A319000, A320322, A321649.

Cf. A325040, A325041, A325042, A325045.

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

More terms from Jinyuan Wang, Jun 27 2020

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).

A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 50, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 315, 320, 324, 336, 352, 360, 375, 378

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than that of their conjugate.

			The terms together with their prime indices begin:
    4: (1,1)
    8: (1,1,1)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   24: (2,1,1,1)
   27: (2,2,2)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
For example, the partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, and 196 > 189, so 196 is in the sequence, and 189 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A000701.
The opposite version is A352487, weak A352489.
The weak version is A352488, counted by A046682.
These are the positions of positive terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = 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.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352521 counts compositions by subdiagonals, rank statistic A352514.
Cf. A000720, A114088, A114324, A321648, A324850, A325037, A325038, A325040.

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],#>Times@@Prime/@conj[primeMS[#]]&]

a(n) > A122111(a(n)).

A329382 Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 4, 1, 6, 1, 3, 4, 2, 1, 4, 8, 2, 9, 3, 1, 6, 1, 5, 4, 2, 8, 8, 1, 2, 4, 4, 1, 6, 1, 3, 9, 2, 1, 5, 16, 12, 4, 3, 1, 12, 8, 4, 4, 2, 1, 8, 1, 2, 9, 6, 8, 6, 1, 3, 4, 12, 1, 10, 1, 2, 18, 3, 16, 6, 1, 5, 16, 2, 1, 8, 8, 2, 4, 4, 1, 12, 16, 3, 4, 2, 8, 6, 1, 24, 9, 16, 1, 6, 1, 4, 18

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Also the product of parts of the conjugate of the integer partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (3,2) with Heinz number 15 has conjugate (2,2,1) with product a(15) = 4. - Gus Wiseman, Mar 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A000005, A005361, A019565, A034386, A108951, A284001, A331188, A329378, A329605, A329617.
This is the conjugate version of A003963 (product of prime indices).
The solutions to a(n) = A003963(n) are A325040, counted by A325039.
The Heinz number of the conjugate partition is given by A122111.
These are the row products of A321649 and of A321650.
A000700 counts self-conj partitions, ranked by A088902, complement A330644.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and of A296150.
A124010 gives prime signature, sorted A118914, sum A001222.
A238744 gives the conjugate of prime signature, rank A238745.
Cf. A000701, A000720, A001221, A046682, A175508, A290822, A303975, A316524, A324850, A352486-A352491, A353570.

Table[Times @@ FactorInteger[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]][[All, -1]], {n, 105}] (* Michael De Vlieger, Jan 21 2020 *)

A005361(n) = factorback(factor(n)[, 2]); \\ from A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329382(n) = A005361(A108951(n));

A329382(n) = if(1==n,1,my(f=factor(n),e=0,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

a(n) = A005361(A108951(n)).
A329605(n) >= a(n) >= A329617(n) >= A329378(n).
a(A019565(n)) = A284001(n).
From Antti Karttunen, Jan 14 2020: (Start)
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > k3 > ... > kx, then a(n) = e(k1)^(k1-k2) * (e(k1)+e(k2))^(k2-k3) * (e(k1)+e(k2)+e(k3))^(k3-k4) * ... * (e(k1)+e(k2)+...+e(kx))^kx.
a(n) = A000005(A331188(n)) = A329605(A052126(n)).
(End)
a(n) = A003963(A122111(n)). - Gus Wiseman, Mar 27 2022

cp's OEIS Frontend

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A352486 Heinz numbers of non-self-conjugate integer partitions.

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A350945 Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

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A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

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A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

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A325039 Number of integer partitions of n with the same product of parts as their conjugate.

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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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