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

Author

Gus Wiseman, Mar 27 2019

Keywords

• nonn

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).

Examples

			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}
		

Crossrefs

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

Programs

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]]]]]&]