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A325042
Heinz numbers of integer partitions whose product of parts is one fewer than their sum.
14
4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 168, 178, 194, 202, 206, 214, 216, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 400, 422, 446, 454, 458, 466
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 one fewer than their sum of prime indices (A056239).
FORMULA
A003963(a(n)) = A056239(a(n)) - 1.
a(n) = 2 * A301987(n).
EXAMPLE
The sequence of terms together with their prime indices begins:
4: {1,1}
6: {1,2}
10: {1,3}
14: {1,4}
18: {1,2,2}
22: {1,5}
26: {1,6}
34: {1,7}
38: {1,8}
46: {1,9}
58: {1,10}
60: {1,1,2,3}
62: {1,11}
74: {1,12}
82: {1,13}
86: {1,14}
94: {1,15}
106: {1,16}
118: {1,17}
122: {1,18}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Times@@primeMS[#]==Total[primeMS[#]]-1&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
STATUS
approved