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A325091
Heinz numbers of integer partitions of powers of 2.
3
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).
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]]]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 27 2019
STATUS
approved