|
|
A348384
|
|
Heinz numbers of integer partitions whose length is 2/3 their sum.
|
|
2
|
|
|
1, 6, 36, 40, 216, 224, 240, 1296, 1344, 1408, 1440, 1600, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 34816, 39936, 46656, 48384, 50176, 50688, 51840, 53760, 56320, 57600, 64000, 155648, 208896, 239616, 266240, 279936, 290304, 301056, 304128, 311040, 315392
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose sum of prime indices is 3/2 their number. Counting the partitions with these Heinz numbers gives A035377(n) = A000041(n/3) if n is a multiple of 3, otherwise 0.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The terms and their prime indices begin:
1: {}
6: {1,2}
36: {1,1,2,2}
40: {1,1,1,3}
216: {1,1,1,2,2,2}
224: {1,1,1,1,1,4}
240: {1,1,1,1,2,3}
1296: {1,1,1,1,2,2,2,2}
1344: {1,1,1,1,1,1,2,4}
1408: {1,1,1,1,1,1,1,5}
1440: {1,1,1,1,1,2,2,3}
1600: {1,1,1,1,1,1,3,3}
6656: {1,1,1,1,1,1,1,1,1,6}
7776: {1,1,1,1,1,2,2,2,2,2}
|
|
MATHEMATICA
|
Select[Range[1000], 2*Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]==3*PrimeOmega[#]&]
|
|
PROG
|
(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
|
|
CROSSREFS
|
These partitions are counted by A035377.
A001222 counts prime factors with multiplicity.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|