A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.

1, 2, 3, 4, 7, 9, 12, 16, 19, 49, 53, 63, 81, 84, 108, 112, 131, 144, 192, 256, 311, 361, 719, 931, 1197, 1539, 1596, 1619, 2052, 2128, 2401, 2736, 2809, 3087, 3648, 3671, 3969, 4116, 4864, 5103, 5292, 5488, 6561, 6804, 7056, 8161, 8748, 9072, 9408, 11664, 12096

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 prime indices are powers of 2 and whose sum of prime indices is also 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..51.

Example

The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    7: {4}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   19: {8}
   49: {4,4}
   53: {16}
   63: {2,2,4}
   81: {2,2,2,2}
   84: {1,1,2,4}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  131: {32}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  311: {64}

Maple

q:= n-> andmap(t-> t=2^ilog2(t), (l-> [l[], add(i, i=l)])(
      map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
select(q, [$1..15000])[];  # Alois P. Heinz, Mar 28 2019

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
pow2Q[n_]:=IntegerQ[Log[2, n]];
Select[Range[1000], #==1||pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]

Crossrefs

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325093.

Sequence in context: A117450 A132381 A073152 * A247186 A051061 A237870

Adjacent sequences: A325089 A325090 A325091 * A325093 A325094 A325095

Keyword

nonn

Author

Gus Wiseman, Mar 27 2019

Status

approved