%I #58 Aug 18 2024 14:10:33

%S 36,100,196,225,441,484,676,1089,1156,1225,1444,1521,2116,2601,3025,

%T 3249,3364,3844,4225,4761,5476,5929,6724,7225,7396,7569,8281,8649,

%U 8836,9025,11236,12321,13225,13924,14161,14884,15129,16641,17689,17956,19881

%N Squares of the squarefree semiprimes (p^2*q^2).

%C This sequence is a member of a family of sequences directly related to A025487. First terms and known sequences are listed below: 1, A000007; 2, A000040; 4, A001248; 6, A006881; 8, A030078; 12, A054753; 16, A030514; 24, A065036; 30, A007304; 32, A050997; 36, this sequence; 48, ?; 60, ?; 64, ?; ....

%C Subsequence of A077448. The numbers in A077448 but not in here are 1, the squares of A046386, the squares of A067885, etc. - _R. J. Mathar_, Sep 12 2008

%C a(4)-a(3)=29 and a(3)+a(4)=421 are both prime. There are no other cases where the sum and difference of two members of this sequence are both prime. - _Robert Israel_ and _J. M. Bergot_, Oct 25 2019

%H T. D. Noe, <a href="/A085986/b085986.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>

%F a(n) = A006881(n)^2.

%F Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 = (A085548^2 - A085964)/2 = 0.063767..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020

%e A006881 begins 6 10 14 15 ... so this sequence begins 36 100 196 225 ...

%t f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2}; Select[Range[20000], f] (* _Vladimir Joseph Stephan Orlovsky_, Aug 14 2009 *)

%t Select[Range[200],PrimeOmega[#]==2&&SquareFreeQ[#]&]^2 (* _Harvey P. Dale_, Mar 07 2013 *)

%o (PARI) list(lim)=my(v=List(), x=sqrtint(lim\=1), t); forprime(p=2, x\2, t=p; forprime(q=2, min(x\t,p-1), listput(v, (t*q)^2))); Set(v) \\ _Charles R Greathouse IV_, Sep 22 2015

%o (PARI) is(n)=factor(n)[,2]==[2,2]~ \\ _Charles R Greathouse IV_, Oct 19 2015

%o (Magma) [k^2:k in [1..150]| IsSquarefree(k) and #PrimeDivisors(k) eq 2]; // _Marius A. Burtea_, Oct 24 2019

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange

%o def A085986(n):

%o def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return m**2 # _Chai Wah Wu_, Aug 18 2024

%Y Subsequence of A036785 and of A077448.

%Y Subsequence of A062503.

%Y Cf. A025487.

%Y Cf. A085548, A085964.

%K easy,nonn

%O 1,1

%A _Alford Arnold_, Jul 06 2003