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A325043
Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer than their sum.
0
18, 60, 168, 216, 400, 528, 1248, 2240, 2880, 3264, 7296, 14080, 17664, 25088, 32256, 41472, 44544, 66560, 95232, 153600, 227328, 315392, 348160, 405504, 503808, 1056768, 1556480, 2310144, 2981888, 3833856, 5210112, 6881280, 7536640, 7929856, 8847360, 11599872
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 with at least three prime factors (counted with multiplicity) whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).
FORMULA
a(n) = 2 * A301988(n).
EXAMPLE
The sequence of terms together with their prime indices begins:
18: {1,2,2}
60: {1,1,2,3}
168: {1,1,1,2,4}
216: {1,1,1,2,2,2}
400: {1,1,1,1,3,3}
528: {1,1,1,1,2,5}
1248: {1,1,1,1,1,2,6}
2240: {1,1,1,1,1,1,3,4}
2880: {1,1,1,1,1,1,2,2,3}
3264: {1,1,1,1,1,1,2,7}
7296: {1,1,1,1,1,1,1,2,8}
14080: {1,1,1,1,1,1,1,1,3,5}
17664: {1,1,1,1,1,1,1,1,2,9}
25088: {1,1,1,1,1,1,1,1,1,4,4}
32256: {1,1,1,1,1,1,1,1,1,2,2,4}
41472: {1,1,1,1,1,1,1,1,1,2,2,2,2}
44544: {1,1,1,1,1,1,1,1,1,2,10}
66560: {1,1,1,1,1,1,1,1,1,1,3,6}
95232: {1,1,1,1,1,1,1,1,1,1,2,11}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], And[PrimeOmega[#]>2, Times@@primeMS[#]==Total[primeMS[#]]-1]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
EXTENSIONS
More terms from Jinyuan Wang, Jun 27 2020
STATUS
approved