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
Keywords
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.
Programs
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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 -
Mathematica
nn = 300; Union[Select[Flatten[Table[p*n^2, {p, Prime[Range[PrimePi[nn/4]]]}, {n, 2, Sqrt[nn/2]}]], # < nn &]] -
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
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Python
from math import isqrt from sympy import primepi def A228056(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 kmin = kmax >> 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): return n+x-sum(primepi(x//y**2) for y in range(2,isqrt(x)+1)) return bisection(f,n,n) # Chai Wah Wu, Jun 06 2025
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
Comments