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A071331
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Numbers having no decomposition into a sum of two prime powers.
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11
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(PARI) 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)
(Haskell)
a071331 n = a071331_list !! (n-1)
a071331_list = filter ((== 0) . a071330) [1..]
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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