A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 17719, 32768, 40807, 43381, 50431, 65536, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 131072, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239

Offset

1,2

Comments

Equals the union A000079 and A328868.

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

A partition with no two distinct parts relatively prime is said to be intersecting.

Links

Table of n, a(n) for n=1..40.

Example

The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     16: {1,1,1,1}
     32: {1,1,1,1,1}
     64: {1,1,1,1,1,1}
    128: {1,1,1,1,1,1,1}
    256: {1,1,1,1,1,1,1,1}
    512: {1,1,1,1,1,1,1,1,1}
   1024: {1,1,1,1,1,1,1,1,1,1}
   2048: {1,1,1,1,1,1,1,1,1,1,1}
   4096: {1,1,1,1,1,1,1,1,1,1,1,1}
   8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
  16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  17719: {6,10,15}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  40807: {6,14,21}
  43381: {6,15,20}
  50431: {10,12,15}
  65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], #==1||GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[primeMS[#]], {2}]]&]

Crossrefs

These are the Heinz numbers of the partitions counted by A328672.

Terms that are not powers of 2 are A328868.

The strict case is A318716.

The version without global relative primality is A328867.

A ranking using binary indices (instead of prime indices) is A326912.

The version for non-isomorphic multiset partitions is A319759.

The version for divisibility (instead of relative primality) is A328677.

Heinz numbers of relatively prime partitions are A289509.

Cf. A000837, A056239, A112798, A200976, A202425, A289509, A291166, A302796, A316476, A318715, A319752, A328336.

Sequence in context: A219676 A220469 A370254 * A220051 A220493 A320487

Adjacent sequences: A328676 A328677 A328678 * A328680 A328681 A328682

Keyword

nonn

Author

Gus Wiseman, Oct 30 2019

Status

approved