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A325041
Heinz numbers of integer partitions whose product of parts is one greater than their sum.
9
1, 15, 42, 54, 100, 132, 312, 560, 720, 816, 1824, 3520, 4416, 6272, 8064, 10368, 11136, 16640, 23808, 38400, 56832, 78848, 87040, 101376, 125952, 264192, 389120, 577536, 745472, 958464, 1302528, 1720320, 1884160, 1982464, 2211840, 2899968, 5996544
OFFSET
1,2
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 more than their sum of prime indices (A056239).
FORMULA
A003963(a(n)) = A056239(a(n)) + 1.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
15: {2,3}
42: {1,2,4}
54: {1,2,2,2}
100: {1,1,3,3}
132: {1,1,2,5}
312: {1,1,1,2,6}
560: {1,1,1,1,3,4}
720: {1,1,1,1,2,2,3}
816: {1,1,1,1,2,7}
1824: {1,1,1,1,1,2,8}
3520: {1,1,1,1,1,1,3,5}
4416: {1,1,1,1,1,1,2,9}
6272: {1,1,1,1,1,1,1,4,4}
8064: {1,1,1,1,1,1,1,2,2,4}
10368: {1,1,1,1,1,1,1,2,2,2,2}
11136: {1,1,1,1,1,1,1,2,10}
16640: {1,1,1,1,1,1,1,1,3,6}
23808: {1,1,1,1,1,1,1,1,2,11}
38400: {1,1,1,1,1,1,1,1,1,2,3,3}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], Times@@primeMS[#]==Total[primeMS[#]]+1&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
STATUS
approved