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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 (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 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).
LINKS
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]&]
CROSSREFS
Sequence in context: A218617 A105521 A154563 * A338536 A090073 A327089
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
EXTENSIONS
More terms from Jinyuan Wang, Jun 27 2020
STATUS
approved

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Last modified February 21 15:08 EST 2024. Contains 370236 sequences. (Running on oeis4.)