A325091 Heinz numbers of integer partitions of powers of 2.

1, 2, 3, 4, 7, 9, 10, 12, 16, 19, 34, 39, 49, 52, 53, 55, 63, 66, 70, 75, 81, 84, 88, 90, 94, 100, 108, 112, 120, 129, 131, 144, 160, 172, 192, 205, 246, 254, 256, 259, 311, 328, 333, 339, 341, 361, 370, 377, 391, 434, 444, 452, 465, 545, 558, 592, 598, 609, 614

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose sum of prime indices is a power of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).

Links

Table of n, a(n) for n=1..59.

Example

The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  16: {1,1,1,1}
  19: {8}
  34: {1,7}
  39: {2,6}
  49: {4,4}
  52: {1,1,6}
  53: {16}
  55: {3,5}
  63: {2,2,4}
  66: {1,2,5}
  70: {1,3,4}
  75: {2,3,3}
  81: {2,2,2,2}

Maple

q:= n-> (t-> t=2^ilog2(t))(add(numtheory[pi](i[1])*i[2], i=ifactors(n)[2])):
select(q, [$1..1000])[];  # Alois P. Heinz, Mar 28 2019

Mathematica

Select[Range[100], #==1||IntegerQ[Log[2, Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]]]&]

Crossrefs

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A131577, A318400, A325092, A325093.

Sequence in context: A050029 A364868 A137451 * A057912 A141748 A057267

Adjacent sequences: A325088 A325089 A325090 * A325092 A325093 A325094

Keyword

nonn

Author

Gus Wiseman, Mar 27 2019

Status

approved