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

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, 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, 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

Offset

1,4

Comments

a(2*n) > 0 (Goldbach's conjecture).

a(A071331(n)) = 0; A095840(n) = a(A000961(n)).

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Example

10 = 1 + 3^2 = 2 + 2^3 = 3 + 7 = 5 + 5, therefore a(10) = 4;
11 = 2 + 3^2 = 3 + 2^3 = 4 + 7, therefore a(11) = 3;
12 = 1 + 11 = 3 + 3^2 = 2^2 + 2^3 = 5 + 7, therefore a(12) = 4;
a(149)=0, as for all x<149: if x is a prime power then 149-x is not.

Mathematica

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 *)

Prog

(PARI) ispp(n) = (omega(n)==1 || n==1)
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
(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
(Haskell)
a071330 n = sum $
   map (a010055 . (n -)) $ takeWhile (<= n `div` 2) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

Crossrefs

Cf. A000961, A002375, A010055, A061358, A071331, A109829.

Sequence in context: A330036 A050430 A277329 * A374758 A358635 A366615

Adjacent sequences: A071327 A071328 A071329 * A071331 A071332 A071333

Keyword

nonn,nice

Author

Reinhard Zumkeller, May 19 2002

Status

approved