A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..33.

Example

The sequence of terms together with their prime indices begins:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
dv=Select[Range[100000], GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]], 2]]&]

Crossrefs

These are the Heinz numbers of the partitions counted by A202425.

Terms of A328679 that are not powers of 2.

The strict case is A318716 (preceded by 2).

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

Heinz numbers of relatively prime partitions are A289509.

Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

Sequence in context: A187641 A252624 A252513 * A322552 A235023 A133540

Adjacent sequences: A328865 A328866 A328867 * A328869 A328870 A328871

Keyword

nonn

Author

Gus Wiseman, Oct 30 2019

Status

approved