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A071331
Numbers having no decomposition into a sum of two prime powers.
11
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
1,2
COMMENTS
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
LINKS
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.
MATHEMATICA
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 *)
PROG
(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)
\\ Charles R Greathouse IV, Jul 06 2011
(Haskell)
a071331 n = a071331_list !! (n-1)
a071331_list = filter ((== 0) . a071330) [1..]
-- Reinhard Zumkeller, Jan 11 2013
CROSSREFS
A071330(a(n))=0. Cf. A000961, A109829, A014092.
Sequence in context: A346891 A244661 A146137 * A095842 A142359 A185692
KEYWORD
nonn,nice
AUTHOR
Reinhard Zumkeller, May 19 2002
STATUS
approved