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A228056
Numbers of the form p * m^2, where p is prime and m > 1.
5
8, 12, 18, 20, 27, 28, 32, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 125, 128, 147, 148, 153, 162, 164, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 242, 243, 244, 245, 252, 261, 268, 272, 275, 279, 284
OFFSET
1,1
COMMENTS
This sequence is the first step toward candidates for odd perfect numbers, A228058.
LINKS
Raghavendra Bhat, Distribution of Square Prime Numbers, arXiv:2109.10238 [math.NT], 2021.
Raghavendra Bhat, An Algebraic Structure for Square-Prime Numbers, arXiv:2303.14296 [math.GM], 2023.
Raghavendra Bhat, Cristian Cobeli, and Alexandru Zaharescu, Filtered rays over iterated absolute differences on layers of integers, arXiv:2309.03922 [math.NT], 2023. See 3.1 p. 9.
FORMULA
Bhat proves there are ~ (Pi^2/6-1)*x/log x members of this sequence up to x, so a(n) ~ kn log n with k = 6/(Pi^2-6) = 1.550546.... - Charles R Greathouse IV, Oct 01 2021
MATHEMATICA
nn = 300; Union[Select[Flatten[Table[p*n^2, {p, Prime[Range[PrimePi[nn/4]]]}, {n, 2, Sqrt[nn/2]}]], # < nn &]]
PROG
(Haskell)
import Data.List (partition)
a228056 n = a228056_list !! (n-1)
a228056_list = filter f [1..] where
f x = length us == 1 && (head us > 1 || not (null vs)) where
(us, vs) = partition odd $ a124010_row x
-- Reinhard Zumkeller, Aug 14 2013
(PARI) list(lim)=my(v=List()); forfactored(n=2, lim\1, my(e=n[2][, 2]); if(vecsum(e%2)==1 && e!=[1]~, listput(v, n[1]))); Vec(v); \\ Charles R Greathouse IV, Oct 01 2021
CROSSREFS
Cf. A124010.
Sequence in context: A046369 A376703 A066428 * A187042 A370650 A285508
KEYWORD
nonn
AUTHOR
T. D. Noe, Aug 13 2013
STATUS
approved