A228056 Numbers of the form p * m^2, where p is prime and m > 1.

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

T. D. Noe, Table of n, a(n) for n = 1..10000

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. A228057, A228058.

Cf. A124010.

Sequence in context: A212582 A046369 A066428 * A187042 A370650 A285508

Adjacent sequences: A228053 A228054 A228055 * A228057 A228058 A228059

Keyword

nonn

Author

T. D. Noe, Aug 13 2013

Status

approved