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A325037
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Heinz numbers of integer partitions whose product of parts is greater than their sum.
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24
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1, 15, 21, 25, 27, 33, 35, 39, 42, 45, 49, 50, 51, 54, 55, 57, 63, 65, 66, 69, 70, 75, 77, 78, 81, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 110, 111, 114, 115, 117, 119, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153
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OFFSET
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1,2
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COMMENTS
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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is greater than their sum of prime indices (A056239).
The enumeration of these partitions by sum is given by A114324.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
15: {2,3}
21: {2,4}
25: {3,3}
27: {2,2,2}
33: {2,5}
35: {3,4}
39: {2,6}
42: {1,2,4}
45: {2,2,3}
49: {4,4}
50: {1,3,3}
51: {2,7}
54: {1,2,2,2}
55: {3,5}
57: {2,8}
63: {2,2,4}
65: {3,6}
66: {1,2,5}
69: {2,9}
70: {1,3,4}
75: {2,3,3}
77: {4,5}
78: {1,2,6}
81: {2,2,2,2}
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MAPLE
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q:= n-> (l-> mul(i, i=l)>add(i, i=l))(map(i->
numtheory[pi](i[1])$i[2], ifactors(n)[2])):
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Times@@primeMS[#]>Plus@@primeMS[#]&]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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