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A154364
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Number of ways to express n as the sum of an odd prime, a positive Pell number and a companion Pell number.
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3
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0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 4, 5, 4, 3, 4, 3, 5, 4, 2, 3, 4, 4, 3, 4, 4, 3, 4, 4, 7, 4, 4, 3, 6, 3, 6, 5, 6, 4, 8, 5, 7
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,10
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COMMENTS
| This is inspired by the sequence A154290 and related conjectures of Sun. On Jan 08, 2009 Zhi-Wei Sun and Qing-Hu Hou conjectured that a(n)>0 for n=6,7,...; in other words, any integer n>5 can be written as the sum of an odd prime, a positive Pell number and a companian Pell number. The Pell numbers are defined by P_0=0, P_1=1 and P_{n+1}=2P_n+P_{n-1} (n=1,2,3,...) and the companion Pell numbers are given by Q_0=Q_1=2 and Q_{n+1}=2Q_n+Q_{n-1} (n=1,2,3...). Note that for n>5 both P_n and Q_n are greater than 2^n. D. McNeil disproved the conjecture by finding the 4 initial counterexamples: 169421772576, 189661491306, 257744272674, 534268276332. On Feb. 1, 2009, Zhi-Wei Sun observed that these 4 counterexamples are divisible by 42 and guessed that all counterexamples to the conjecture of Sun and Hou should be multiples of 42.
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REFERENCES
| R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
Z. W. Sun, A congruence for primes, Proc. Amer. Math. Soc. 123(1995), 1341-1346.
T. Tao, A remark on primality testing and decimal expansions, J. Austral. Math. Soc., in press. arXiv:0802.3361
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LINKS
| Zhi-Wei SUN, Table of n, a(n), n=1..100000.
D. S. McNeil, Sun's strong conjecture
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
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EXAMPLE
| For n=10 the a(10)=3 solutions are 3+5+2, 3+1+6, 7+1+2.
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MAPLE
| Pell:=proc(n) if n=0 then return(0); elif n=1 then return(1); else return( 2*Pell(n-1) + Pell(n-2) ); fi; end proc: comp_Pell:=proc(n) if n=0 then return(2); elif n=1 then return(2); else return( 2*comp_Pell(n-1) + comp_Pell(n-2) ); fi; end proc: for n from 1 to 10^5 do rep_num:=0; for i from 1 while Pell(i)<n do for j from 1 while Pell(i)+comp_Pell(j)<n do p:=n-Pell(i)-comp_Pell(j); if (p>2) and isprime(p) then rep_num:=rep_num+1; fi; od; od; printf("%d %d\n", n, rep_num); od:
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CROSSREFS
| Cf. A000040, A000129, A002203, A154290, A154257, A154285
Sequence in context: A068448 A054081 A164585 * A183043 A050604 A131015
Adjacent sequences: A154361 A154362 A154363 * A154365 A154366 A154367
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KEYWORD
| nice,nonn
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AUTHOR
| Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 07 2009
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EXTENSIONS
| D. McNeil has disproved the conjecture. ---Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 17 2009
Added Sun's observation that known counterexamples are all divisible by 42 Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Feb 01 2009
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