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A325092
Heinz numbers of integer partitions of powers of 2 into powers of 2.
3
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).
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[#]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 27 2019
STATUS
approved