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A071330 Number of decompositions of n into sum of two prime powers.

%I A071330 #27 Mar 07 2018 13:23:58
%S A071330 0,1,1,2,2,3,2,3,3,4,3,4,3,4,3,4,3,5,3,5,4,4,2,5,3,5,4,5,3,6,3,7,5,7,
%T A071330 4,7,2,6,4,6,3,6,3,6,5,6,2,8,3,8,4,6,2,9,3,7,4,6,2,8,3,7,4,7,3,9,2,8,
%U A071330 5,7,2,10,3,8,6,7,3,9,2,9,4,7,4,11,3,9,4,7,3,12,4,8,3,7,2
%N A071330 Number of decompositions of n into sum of two prime powers.
%C A071330 a(2*n) > 0 (Goldbach's conjecture).
%C A071330 a(A071331(n)) = 0; A095840(n) = a(A000961(n)).
%H A071330 T. D. Noe, <a href="/A071330/b071330.txt">Table of n, a(n) for n = 1..10000</a>
%e A071330 10 = 1 + 3^2 = 2 + 2^3 = 3 + 7 = 5 + 5, therefore a(10) = 4;
%e A071330 11 = 2 + 3^2 = 3 + 2^3 = 4 + 7, therefore a(11) = 3;
%e A071330 12 = 1 + 11 = 3 + 3^2 = 2^2 + 2^3 = 5 + 7, therefore a(12) = 4;
%e A071330 a(149)=0, as for all x<149: if x is a prime power then 149-x is not.
%t A071330 primePowerQ[n_] := Length[ FactorInteger[n]] == 1; a[n_] := (r = 0; Do[ If[ primePowerQ[k] && primePowerQ[n-k], r++], {k, 1, Floor[n/2]}]; r); Table[a[n], {n, 1, 95}](* _Jean-François Alcover_, Nov 17 2011, after _Michael B. Porter_ *)
%o A071330 (PARI) ispp(n) = (omega(n)==1 || n==1)
%o A071330 A071330(n) = {local(r);r=0;for(i=1,floor(n/2),if(ispp(i) && ispp(n-i),r++));r} \\ _Michael B. Porter_, Dec 04 2009
%o A071330 (PARI) a(n)=my(s); forprime(p=2,n\2,if(isprimepower(n-p), s++)); for(e=2,log(n)\log(2), forprime(p=2, sqrtnint(n\2,e), if(isprimepower(n-p^e), s++))); s+(!!isprimepower(n-1))+(n==2) \\ _Charles R Greathouse IV_, Nov 21 2014
%o A071330 (Haskell)
%o A071330 a071330 n = sum $
%o A071330    map (a010055 . (n -)) $ takeWhile (<= n `div` 2) a000961_list
%o A071330 -- _Reinhard Zumkeller_, Jan 11 2013
%Y A071330 Cf. A000961, A002375, A010055, A061358, A071331, A109829.
%K A071330 nonn,nice
%O A071330 1,4
%A A071330 _Reinhard Zumkeller_, May 19 2002