A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

27, 45, 54, 63, 75, 81, 90, 99, 105, 108, 117, 126, 135, 147, 150, 153, 162, 165, 171, 180, 189, 195, 198, 207, 210, 216, 225, 231, 234, 243, 252, 255, 261, 270, 273, 279, 285, 294, 297, 300, 306, 315, 324, 330, 333, 342, 345, 351, 357, 360, 363, 369, 378, 387

Offset

1,1

Comments

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 A006918 (see Emeric Deutsch's comment there).

Links

Table of n, a(n) for n=1..54.

Example

The sequence of terms together with their prime indices begins:
   27: {2,2,2}
   45: {2,2,3}
   54: {1,2,2,2}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   90: {1,2,2,3}
   99: {2,2,5}
  105: {2,3,4}
  108: {1,1,2,2,2}
  117: {2,2,6}
  126: {1,2,2,4}
  135: {2,2,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  153: {2,2,7}
  162: {1,2,2,2,2}
  165: {2,3,5}
  171: {2,2,8}
  180: {1,1,2,2,3}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], PrimeOmega[#]>=3&&Reverse[primeMS[#]][[3]]==2&]

Crossrefs

Cf. A000726, A002620, A004250, A006918, A056239, A097701, A112798, A257990, A297113, A325164, A325169, A325170.

Sequence in context: A253919 A357077 A259504 * A121614 A046340 A046316

Adjacent sequences: A307370 A307371 A307372 * A307374 A307375 A307376

Keyword

nonn

Author

Gus Wiseman, Apr 05 2019

Status

approved