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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 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
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[#]&]
CROSSREFS
Sequence in context: A070776 A320230 A076564 * A343822 A303550 A164563
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
STATUS
approved

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Last modified February 25 21:07 EST 2024. Contains 370333 sequences. (Running on oeis4.)