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A325044
Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.
25
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 88, 89, 92, 94, 96, 97, 101
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 or equal to their sum of prime indices (A056239).
The enumeration of these partitions by sum is given by A096276.
FORMULA
A003963(a(n)) <= A056239(a(n)).
a(n) = A325038(n)/2.
Union of A301987 and A325038.
EXAMPLE
The sequence of terms together with their prime indices begins:
2: {1}
3: {2}
4: {1,1}
5: {3}
6: {1,2}
7: {4}
8: {1,1,1}
9: {2,2}
10: {1,3}
11: {5}
12: {1,1,2}
13: {6}
14: {1,4}
16: {1,1,1,1}
17: {7}
18: {1,2,2}
19: {8}
20: {1,1,3}
22: {1,5}
23: {9}
24: {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