

A361855


Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).


14



28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254
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OFFSET

1,1


COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also positive integers whose prime indices satisfy (maximum) = 2*(mean).
Also Heinz numbers of partitions of the same size as their complement (see example).


LINKS



FORMULA



EXAMPLE

The terms together with their prime indices begin:
28: {1,1,4}
40: {1,1,1,3}
78: {1,2,6}
84: {1,1,2,4}
171: {2,2,8}
190: {1,3,8}
198: {1,2,2,5}
220: {1,1,3,5}
240: {1,1,1,1,2,3}
252: {1,1,2,2,4}
280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
o o o o o
o o . . .
o o . . .
o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.


MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], Max@@prix[#]*PrimeOmega[#]==2*Total[prix[#]]&]


CROSSREFS

A061395 gives greatest prime index.


KEYWORD

nonn


AUTHOR



STATUS

approved



