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%I #15 Jun 02 2015 15:57:59
%S 1,4,13,49,161,656,2751,12505,58482,280348,1374563,6864809
%N Number of Goldbach partitions of 6^n.
%C Number of ways of writing 6^n as the sum of two odd primes, when the order does not matter. Number of ways writing 6^n as unordered sums of 2 primes. This is to 6 as A006307 is to 2 and as A065577 is to 10. This is the 6th row of the array A[k,n] = Number of ways writing k^n as unordered sums of 2 primes.
%C A061358(4^n) starts 1, 2, 5, 8, 22, 53, 151, 435, for n=1,2,... (bisection of A006307). A061358(8^n) starts 1, 5, 11, 53, 244, 1314, 7471, (tri-section of A006307). A061358(10^n) = A065577(n). A061358(12^n) = 1, 11, 53, 348, 2523, 20564... A061358(14^n) = 2, 9, 50, 330, 2924, 27225,... - _R. J. Mathar_, Aug 07 2010
%H Manfred Scheucher, <a href="/A180007/a180007.sage.txt">Sage Script</a>
%F a(n) = A061358(6^n) = A061358(A000400(n)).
%e a(1) = 1 because 6^1 = 6 = 3+3.
%e a(2) = 4 because 6^2 = 36 = 5+31 = 7+29 = 13+23 = 17+19.
%e a(3) = 13 because 6^3 = 216 = 5+211 = 17+199 = 19+197 = 23+193 = 37+179 = 43+173 = 53+163 = 59+157 = 67+149 = 79+137 = 89+127 = 103+113 = 107+109.
%p A061358 := proc(n) local a,p ; a := 0 ; p := nextprime(floor((n-1)/2)) ; while p <= n do if isprime(n-p) then a := a+1 ; end if; p := nextprime(p) ; end do ; return a; end proc:
%p A180007 := proc(n) A061358(6^n) ; end proc:
%p for n from 1 do printf("%d,\n",A180007(n)) ; end do:
%p # _R. J. Mathar_, Aug 07 2010
%t Table[Count[Sort@ IntegerPartitions[6^n, {2}], {u_, v_} /; And[PrimeQ@ u, u != 2, PrimeQ@ v]], {n, 6}] (* _Michael De Vlieger_, Jun 02 2015 *)
%o (PARI) a(n)=my(t=6^n,s); forprime(p=2,t\2, if(isprime(t-p), s++)); s \\ _Charles R Greathouse IV_, Jun 02 2015
%Y Cf. A000400, A001031, A061358, A065577, A195295.
%K nonn
%O 1,2
%A _Jonathan Vos Post_, Aug 06 2010
%E a(5) corrected, 4 terms added by _R. J. Mathar_, Aug 07 2010
%E a(10)-a(12) from _Manfred Scheucher_, Jun 01 2015
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