A258119 -id:A258119 - cp's OEIS frontend

A325780 Heinz numbers of perfect integer partitions.

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878

Offset: 1

Gus Wiseman, May 21 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     16: {1,1,1,1}
     18: {1,2,2}
     20: {1,1,3}
     32: {1,1,1,1,1}
     42: {1,2,4}
     54: {1,2,2,2}
     56: {1,1,1,4}
     64: {1,1,1,1,1,1}
    100: {1,1,3,3}
    128: {1,1,1,1,1,1,1}
    162: {1,2,2,2,2}
    176: {1,1,1,1,5}
    234: {1,2,2,6}
    256: {1,1,1,1,1,1,1,1}
    260: {1,1,3,6}

Equals the sorted concatenation of the triangle A258119.
A subsequence of A299702 and A325781.
Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]

Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).

A258118 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all complete partitions of n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 30, 36, 40, 48, 64, 42, 54, 56, 60, 72, 80, 96, 128, 84, 90, 100, 108, 112, 120, 144, 160, 192, 256, 126, 132, 140, 150, 162, 168, 176, 180, 200, 216, 224, 240, 288, 320, 384, 512, 198, 210, 220, 252, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 576, 640, 768, 1024

Offset: 0

Emeric Deutsch, Jun 07 2015

A partition of n is complete if every number from 1 to n can be represented as a sum of parts of the partition.
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (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 because the partition [1,1,1,4] is complete.
Except for a(0)=1, there are no odd numbers in the sequence. Indeed, a partition having an odd Heinz number does not have 1 as a part and, consequently, it cannot be complete.
Number of terms in row n is A126796(n). As a matter of fact, so far, the triangle has been constructed by selecting those A126796(n) entries from row n of A215366 which correspond to complete partitions. Last term in row n is 2^n.

			54 = 2*3*3*3 is in the sequence because the partition [1,2,2,2] is complete.
28 = 2*2*7 is not in the sequence because the partition [1,1,4] is not complete.
Triangle T(n,k) begins:
   1;
   2;
   4;
   6,  8;
  12, 16;
  18, 20,  24,  32;
  30, 36,  40,  48,  64;
  42, 54,  56,  60,  72,  80,  96, 128;
  84, 90, 100, 108, 112, 120, 144, 160, 192, 256;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

Cf. A000079, A215366, A126796, A258119.
Column k=1 gives A259941.
Row sums give A360791.

T:= proc(m) local b, ll, p;
      p:= proc(l) ll:=ll, (mul(ithprime(j), j=l)); 1 end:
      b:= proc(n, i, l) `if`(i<2, p([l[], 1$n]), `if`(n<2*i-1,
      b(n, iquo(n+1, 2), l), b(n, i-1, l)+b(n-i, i, [l[], i])))
      end: ll:= NULL; b(m, iquo(m+1, 2), []): sort([ll])[]
    end:
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 07 2015

T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, Product[Prime[j], {j, l}]]; 1); b[n_, i_, l_List] := If[i<2, p[Join[l, Array[1&, n]]], If[n < 2*i-1, b[n, Quotient[n+1, 2], l], b[n, i-1, l] + b[n-i, i, Append[l, i] ]]]; ll = {}; b[m, Quotient[m+1, 2], {}]; Sort[ll]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)

A259939 Smallest Product_{i:lambda} prime(i) for any perfect partition lambda of n.

Original entry on oeis.org

1, 2, 4, 6, 16, 18, 64, 42, 100, 162, 1024, 234, 4096, 1088, 1936, 798, 65536, 2300, 262144, 4698, 18496, 31744, 4194304, 8658, 234256, 167936, 52900, 46784, 268435456, 90992, 1073741824, 42294, 984064, 3866624, 5345344, 140300, 68719476736, 17563648, 6885376

Offset: 0

Alois P. Heinz, Jul 09 2015

nonn

A perfect partition of n contains a unique partition for any k in {0,...,n}. See also A002033.

			For n=7 there are 4 perfect partitions: [4,1,1,1], [4,2,1], [2,2,2,1] and [1,1,1,1,1,1,1], their encodings as Product_{i:lambda} prime(i) give 56, 42, 54, 128, respectively.  The smallest value is a(7) = 42.

Alois P. Heinz, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Perfect Partition

Column k=1 of A258119.

Cf. A002033, A215366, A259941.

b:= (n, l)-> `if`(n=1, 2^(l[1]-1)*mul(ithprime(mul(l[j],
      j=1..i-1))^(l[i]-1), i=2..nops(l)), min(seq(b(n/d,
        [l[], d]), d=numtheory[divisors](n) minus{1}))):
a:= n-> `if`(n=0, 1, b(n+1, [])):
seq(a(n), n=0..42);

b[n_, l_] := If[n==1, 2^(l[[1]]-1)*Product[Prime[Product[l[[j]], {j, 1, i-1}]]^(l[[i]]-1), {i, 2, Length[l]}], Min[Table[b[n/d, Append[l, d]], {d, Divisors[n] ~Complement~ {1}}]]];
a[n_] := If[n==0, 1, b[n+1, {}]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

a(n) = A258119(n,1).

A360713 Sum of all prime encoded perfect partitions of n.

Original entry on oeis.org

1, 2, 4, 14, 16, 70, 64, 280, 356, 850, 1024, 4630, 4096, 10738, 20820, 47264, 65536, 176712, 262144, 643214, 1129572, 2246994, 4194304, 9716880, 17011472, 34785250, 68859688, 139829626, 268435456, 560518826, 1073741824, 2192136576, 4335013860, 8679894658

Offset: 0

Alois P. Heinz, Feb 21 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..3321

Row sums of A258119.

Cf. A002033, A215366, A360791.

b:= proc(n, l) option remember; `if`(n=1,
      mul(ithprime(mul(l[j], j=1..i-1))^(l[i]-1), i=1..nops(l)),
      add(b(n/d, [l[], d]), d=numtheory[divisors](n) minus{1}))
    end:
a:= n-> b(n+1, []):
seq(a(n), n=0..33);

b[n_, l_] := b[n, l] = If[n == 1, Product[Prime[Product[l[[j]], {j, 1, i - 1}]]^(l[[i]] - 1), {i, 1, Length[l]}], Sum[b[n/d, Append[l, d]], {d, Divisors[n]~Complement~{1}}]];
a[n_] := b[n + 1, {}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Nov 21 2023, after Alois P. Heinz *)

a(n) = Sum_{k=1..A002033(n)} A258119(n,k).

cp's OEIS Frontend

A325780 Heinz numbers of perfect integer partitions.

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A258118 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all complete partitions of n.

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A259939 Smallest Product_{i:lambda} prime(i) for any perfect partition lambda of n.

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A360713 Sum of all prime encoded perfect partitions of n.

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