cp's OEIS Frontend

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

%I A228056 #28 Jul 27 2025 10:32:13
%S A228056 8,12,18,20,27,28,32,44,45,48,50,52,63,68,72,75,76,80,92,98,99,108,
%T A228056 112,116,117,124,125,128,147,148,153,162,164,171,172,175,176,180,188,
%U A228056 192,200,207,208,212,236,242,243,244,245,252,261,268,272,275,279,284
%N A228056 Numbers of the form p * m^2, where p is prime and m > 1.
%C A228056 This sequence is the first step toward candidates for odd perfect numbers, A228058.
%H A228056 T. D. Noe, <a href="/A228056/b228056.txt">Table of n, a(n) for n = 1..10000</a>
%H A228056 Raghavendra Bhat, <a href="https://arxiv.org/abs/2109.10238">Distribution of Square Prime Numbers</a>, arXiv:2109.10238 [math.NT], 2021.
%H A228056 Raghavendra Bhat, <a href="https://arxiv.org/abs/2303.14296">An Algebraic Structure for Square-Prime Numbers</a>, arXiv:2303.14296 [math.GM], 2023.
%H A228056 Raghavendra Bhat, Cristian Cobeli, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2309.03922">Filtered rays over iterated absolute differences on layers of integers</a>, arXiv:2309.03922 [math.NT], 2023. See 3.1 p. 9.
%F A228056 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
%t A228056 nn = 300; Union[Select[Flatten[Table[p*n^2, {p, Prime[Range[PrimePi[nn/4]]]}, {n, 2, Sqrt[nn/2]}]], # < nn &]]
%o A228056 (Haskell)
%o A228056 import Data.List (partition)
%o A228056 a228056 n = a228056_list !! (n-1)
%o A228056 a228056_list = filter f [1..] where
%o A228056    f x = length us == 1 && (head us > 1 || not (null vs)) where
%o A228056          (us,vs) = partition odd $ a124010_row x
%o A228056 -- _Reinhard Zumkeller_, Aug 14 2013
%o A228056 (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
%o A228056 (Python)
%o A228056 from math import isqrt
%o A228056 from sympy import primepi
%o A228056 def A228056(n):
%o A228056     def bisection(f,kmin=0,kmax=1):
%o A228056         while f(kmax) > kmax: kmax <<= 1
%o A228056         kmin = kmax >> 1
%o A228056         while kmax-kmin > 1:
%o A228056             kmid = kmax+kmin>>1
%o A228056             if f(kmid) <= kmid:
%o A228056                 kmax = kmid
%o A228056             else:
%o A228056                 kmin = kmid
%o A228056         return kmax
%o A228056     def f(x): return n+x-sum(primepi(x//y**2) for y in range(2,isqrt(x)+1))
%o A228056     return bisection(f,n,n) # _Chai Wah Wu_, Jun 06 2025
%Y A228056 Cf. A228057, A228058.
%Y A228056 Cf. A124010.
%K A228056 nonn
%O A228056 1,1
%A A228056 _T. D. Noe_, Aug 13 2013