proposed

approved

editing

proposed

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

1,12

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

1: {}

Select[Range[200001000], #==1||pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]

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

approved

editing

proposed

approved

editing

proposed

allocated for Gus WisemanHeinz numbers of integer partitions of powers of 2 into powers of 2.

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

1,1

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.

The sequence of terms together with their prime indices begins:

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}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

pow2Q[n_]:=IntegerQ[Log[2, n]];

Select[Range[20000], pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]

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

allocated

nonn

Gus Wiseman, Mar 27 2019

approved

editing

allocated for Gus Wiseman

allocated

approved