A154364 Number of ways to express n as the sum of an odd prime, a positive Pell number and a companion Pell number.

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, 4, 5, 3, 7, 5, 5, 5, 5, 4, 5, 3, 6, 4, 4, 4, 7, 4, 6, 4, 4, 2, 6, 3, 7, 6, 6, 6, 7, 6, 6, 3, 7, 6, 3, 4, 9, 9

Offset

1,10

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. S. McNeil disproved the conjecture by finding the 4 initial counterexamples: 169421772576, 189661491306, 257744272674, 534268276332. - Zhi-Wei Sun, Jan 17 2009

On Feb 01 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. - Zhi-Wei Sun, Feb 01 2009

Links

Zhi-Wei Sun, Table of n, a(n), n=1..100000.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

D. S. McNeil, Sun's strong conjecture, NMBRTHRY, Dec 2008.

Z. W. Sun, A congruence for primes, Proc. Amer. Math. Soc. 123(1995), 1341-1346.

Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t, NMBRTHRY, Dec 2008.

Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t, NMBRTHRY, Dec 2008.

Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010.

Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Example

For n=10 the a(10)=3 solutions are 3+5+2, 3+1+6, 7+1+2.

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:

Mathematica

nmax = 10^3;
Pell[n_] := Pell[n] = If[n == 0, Return[0], If[n == 1, Return[1], Return[2* Pell[n - 1] + Pell[n - 2]]]];
compPell[n_] := compPell[n] = If[n == 0, Return[2], If[n == 1, Return[2],  Return[2*compPell[n - 1] + compPell[n - 2]]]];
Reap[For[n = 1, n <= nmax, n++, repnum = 0; For[i = 1, Pell[i] < n, i++, For[j = 1, Pell[i] + compPell[j] < n, j++, p = n - Pell[i] - compPell[j]; If[p > 2 && PrimeQ[p], repnum++]]]; Sow[repnum]]][[2, 1]] (* Jean-François Alcover, Dec 13 2017, translated from Maple *)

Crossrefs

Cf. A000040, A000129, A002203, A154290, A154257, A154285, A156695.

Sequence in context: A054081 A164585 A200996 * A183043 A050604 A262672

Adjacent sequences: A154361 A154362 A154363 * A154365 A154366 A154367

Keyword

nonn

Author

Zhi-Wei Sun, Jan 07 2009

Status

approved