A109829 -id:A109829 - cp's OEIS frontend

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

Reinhard Zumkeller, May 19 2002

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

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

			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.

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

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

a071330 n = sum $
   map (a010055 . (n -)) $ takeWhile (<= n `div` 2) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

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 *)

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

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

A071331 Numbers having no decomposition into a sum of two prime powers.

Original entry on oeis.org

1, 149, 331, 373, 509, 701, 757, 809, 877, 907, 959, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, 1529, 1541, 1549, 1589, 1597, 1619, 1657, 1719, 1759, 1777, 1783, 1807, 1829, 1859, 1867, 1927, 1969, 1973, 2171, 2231

Offset: 1

Reinhard Zumkeller, May 19 2002

Luca & Stanica show that this sequence contains infinitely many Fibonacci numbers. In particular, there is some N such that for all n > N, Fibonacci(1807873 + 3543120*n) is in this sequence. - Charles R Greathouse IV, Jul 06 2011

Chen shows that there are five consecutive odd numbers M-8, M-6, M-4, M-2, M, for which all are members of the sequence. Such M may be large; Chen shows that it is less than 2^(2^253000). In fact, there exists an arithmetic progression of such M, and thus they have positive density. - Charles R Greathouse IV, Jul 06 2011

T. D. Noe, Table of n, a(n) for n = 1..1000
Florian Luca and Pantelimon Stănică, Fibonacci numbers that are not sums of two prime powers, Proceedings of the American Mathematical Society 133 (2005), pp. 1887-1890.
Yong-Gao Chen, Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers, Mathematics of Computation 74 (2005), pp. 1025-1031.

A071330(a(n))=0. Cf. A000961, A109829, A014092.

a071331 n = a071331_list !! (n-1)
a071331_list = filter ((== 0) . a071330) [1..]
-- Reinhard Zumkeller, Jan 11 2013

primePowerQ[n_] := Length[FactorInteger[n]] == 1; decomposableQ[n_] := (r = False; Do[If[primePowerQ[k] && primePowerQ[n - k], r = True; Break[]], {k, 1, Floor[n/2]}]; r); Select[Range[3000], !decomposableQ[#]& ] (* Jean-François Alcover, Jun 13 2012 *)
Join[{1},Select[Range[4,2300],Count[IntegerPartitions[#,{2}],?( AllTrue[ #,PrimePowerQ]&)]==0&]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, May 28 2021 *)

isprimepower(n)=ispower(n,,&n);isprime(n)||n==1;
isA071331(n)=forprime(p=2,n\2,if(isprimepower(n-p),return(0)));forprime(p=2,sqrtint(n\2),for(e=1,log(n\2)\log(p),if(isprimepower(n-p^e),return(0))));!isprimepower(n-1)
\\ Charles R Greathouse IV, Jul 06 2011

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

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A071331 Numbers having no decomposition into a sum of two prime powers.

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