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A325038
Heinz numbers of integer partitions whose sum of parts is greater than their product.
15
4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172
OFFSET
1,1
COMMENTS
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 less than their sum of prime indices (A056239).
The enumeration of these partitions by sum is given by A096276 shifted once to the right.
FORMULA
A003963(a(n)) < A056239(a(n)).
a(n) = 2 * A325044(n).
EXAMPLE
The sequence of terms together with their prime indices begins:
4: {1,1}
6: {1,2}
8: {1,1,1}
10: {1,3}
12: {1,1,2}
14: {1,4}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
26: {1,6}
28: {1,1,4}
32: {1,1,1,1,1}
34: {1,7}
36: {1,1,2,2}
38: {1,8}
40: {1,1,1,3}
44: {1,1,5}
46: {1,9}
48: {1,1,1,1,2}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Times@@primeMS[#]<Plus@@primeMS[#]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
STATUS
approved