A325045 -id:A325045 - cp's OEIS frontend

A325040 Heinz numbers of integer partitions with the same product of parts as their conjugate.

1, 2, 6, 9, 20, 30, 49, 56, 70, 75, 81, 84, 90, 125, 176, 182, 210, 264, 350, 416, 441, 532, 540, 546, 624, 660, 735, 910, 1088, 1100, 1260, 1378, 1386, 1443, 1520, 1560, 1624, 1632, 1715, 2100, 2310, 2401, 2405, 2432, 2489, 2600, 3024, 3267, 3276, 3648, 3744

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

For example, 182 is the Heinz number of (6,4,1) with product 24 and conjugate (3,2,2,2,1,1) with product also 24.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k).
The enumeration of these partitions by sum is given by A325039.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   20: {1,1,3}
   30: {1,2,3}
   49: {4,4}
   56: {1,1,1,4}
   70: {1,3,4}
   75: {2,3,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
  125: {3,3,3}
  176: {1,1,1,1,5}
  182: {1,4,6}
  210: {1,2,3,4}
  264: {1,1,1,2,5}
  350: {1,3,3,4}
  416: {1,1,1,1,1,6}

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A122111, A321650.

Cf. A325039, A325045.

priptn[n_]:=If[n==1,{},Reverse[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@@priptn[#]==Times@@conj[priptn[#]]&]

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

A353645 a(1) = 1, and for n > 1, number of ways to write the square of n as a product of at least two factors, the largest two of which are equal.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 3, 1, 5, 1, 3, 2, 7, 1, 5, 1, 6, 2, 3, 1, 10, 2, 3, 4, 6, 1, 8, 1, 12, 3, 3, 2, 15, 1, 3, 3, 12, 1, 9, 1, 7, 5, 3, 1, 21, 2, 6, 3, 7, 1, 13, 2, 12, 3, 3, 1, 21, 1, 3, 5, 21, 2, 11, 1, 8, 3, 7, 1, 35, 1, 3, 5, 8, 2, 12, 1, 23, 7, 3, 1, 25, 2, 3, 3, 15, 1, 21, 2, 8, 3, 3, 2, 43, 1, 6, 6, 16

Offset: 1

Antti Karttunen, May 03 2022

nonn

			For n=6, 6^2 = 36 can be factorized in two ways so that two largest factors are equal, as 2*2*3*3 = 6*6, therefore a(6) = 2. See also the examples in A325045.

Antti Karttunen, Table of n, a(n) for n = 1..16415

Cf. A000290, A325045.

A325045(n, m=n, facs=List([])) = if(1==n, (0==#facs || (#facs>=2 && facs[1]==facs[2])), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A325045(n/d, d, newfacs))); (s));
A353645(n) = A325045(n^2);

a(n) = A325045(A000290(n)).

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A325040 Heinz numbers of integer partitions with the same product of parts as their conjugate.

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A353645 a(1) = 1, and for n > 1, number of ways to write the square of n as a product of at least two factors, the largest two of which are equal.

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