

A085986


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


41



36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129, 16641, 17689, 17956, 19881
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OFFSET

1,1


COMMENTS

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, ?; ....
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
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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
Index to sequences related to prime signature


FORMULA

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


EXAMPLE

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


MAPLE

with(numtheory): P:=proc(n)
if nops(factorset(n))=2 and bigomega(n)=2 then n^2; fi;
end: seq(P(i), i=1..141); # Paolo P. Lava, Jul 18 2019


MATHEMATICA

f[n_]:=Sort[Last/@FactorInteger[n]]=={2, 2}; Select[Range[20000], f] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2009 *)
Select[Range[200], PrimeOmega[#]==2&&SquareFreeQ[#]&]^2 (* Harvey P. Dale, Mar 07 2013 *)


PROG

(PARI) list(lim)=my(v=List(), x=sqrtint(lim\=1), t); forprime(p=2, x\2, t=p; forprime(q=2, min(x\t, p1), listput(v, (t*q)^2))); Set(v) \\ Charles R Greathouse IV, Sep 22 2015
(PARI) is(n)=factor(n)[, 2]==[2, 2]~ \\ Charles R Greathouse IV, Oct 19 2015
(Magma) [k^2:k in [1..150] IsSquarefree(k) and #PrimeDivisors(k) eq 2]; // Marius A. Burtea, Oct 24 2019


CROSSREFS

Subsequence of A036785 and of A077448.
Subsequence of A062503.
Cf. A025487.
Cf. A085548, A085964.
Sequence in context: A030627 A177492 A077448 * A027603 A250813 A268770
Adjacent sequences: A085983 A085984 A085985 * A085987 A085988 A085989


KEYWORD

easy,nonn


AUTHOR

Alford Arnold, Jul 06 2003


STATUS

approved



