A154258 -id:A154258 - cp's OEIS frontend

A154285 Number of ordered triples satisfying p+L_s+L_t=n, where p is an odd prime, s and t are nonnegative and the Lucas number L_s or L_t is odd.

0, 0, 0, 0, 1, 2, 3, 6, 4, 8, 5, 8, 6, 8, 9, 12, 10, 12, 9, 10, 12, 14, 9, 14, 12, 14, 10, 14, 8, 10, 10, 16, 11, 16, 12, 18, 12, 16, 10, 12, 13, 16, 15, 16, 13, 14, 13, 16, 14, 18

Offset: 1

Zhi-Wei Sun, Jan 06 2009

nonn

Zhi-Wei Sun conjectured that a(n)>0 for all n=5,6,...; in other words, any integer n>4 can be represented as the sum of an odd prime, an odd Lucas number and a Lucas number. This has been verified up to 1.5*10^8. Sun thought that the constant lim inf_n a(n)/log(n) is greater than 2 and smaller than 3. For k=2,3 Sun also conjectured that any integer n>4 can be written in the form p+L_s+(L_t)^k, where p is an odd prime and L_s or L_t is odd.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

			For n=3 the a(7)=4 solutions are 3+L_1+L_2, 3+L_2+L_1, 5+L_1+L_1.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

Zhi-Wei SUN, Table of n, a(n), n=1..50000.
D. S. McNeil, Sun's strong conjecture
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t (II)
K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp. 78 (2009) 1853, arXiv:math.NT/0702382

Cf. A000032, A000040, A154257, A154258, A154263, A156695.

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n,2]==0||Mod[x,3]>0)&&PQ[n-(2*Fibonacci[x+1]-Fibonacci[x])-(2*Fibonacci[y+1]-Fibonacci[y])],1,0], {x,0,2*Log[2,n]},{y,0,2*Log[2,Max[1,n-(2*Fibonacci[x+1]-Fibonacci[x])]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]

A155114 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 2, 6, 3, 7, 3, 8, 5, 8, 6, 10, 5, 11, 6, 13, 7, 13, 7, 14, 5, 14, 7, 15, 8, 14, 4, 18, 8, 17, 7, 15, 5, 15, 11, 16, 8, 15, 7, 17, 12, 19, 10, 20, 10, 17, 10, 17, 13, 15, 11, 18, 8, 20, 10, 17, 9, 18, 11, 21, 11, 21, 7, 20, 11, 18, 11, 22, 9, 25, 11, 24, 13, 19, 14, 20, 11

Offset: 1

Zhi-Wei Sun, Jan 20 2009

Motivated by his conjecture related to A154257, on Dec 26 2008, Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,... On Jan 15 2009, D. S. McNeil verified this up to 10^12 and found no counterexamples. See the sequence A154536 for another conjecture of this sort. Sun also conjectured that any integer n>7 can be written as the sum of an odd prime, twice a positive Fibonacci number and the square of a positive Fibonacci number; this has been verified up to 2*10^8.

			For n=10 the a(10)=6 solutions are 3 + F_4 + 2F_3, 3 + F_5 + 2F_2, 3 + F_2 + 2F_4, 5 + F_2 + 2F_3, 5 + F_4 + 2F_2, 7 + F_2 + 2F_2.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
D. S. McNeil, Various and sundry (a report on Sun's conjectures)
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 (II)
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp. 78 (2009) 1853-1866. arXiv:math.NT/0702382.

Cf. A000040, A000045, A154257, A154258, A154263, A154285, A154290, A154417, A154536, A154404, A154940, A156695.

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[x]-Fibonacci[y]],1,0], {x,2,2*Log[2,Max[2,n/2]]},{y,2,2*Log[2,Max[2,n-2*Fibonacci[x]]]}] Do[Print[n," ",RN[n]];Continue,{n,1,100000}]

a(n) = |{: p+F_s+2F_t=n with p an odd prime and s,t>1}|.

A154263 Number of triples such that p+F_s+(F_t)^3=n, where p is an odd prime, s and t are greater than one and F_s or F_t is odd.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 3, 1, 2, 2, 2, 4, 2, 5, 3, 2, 3, 4, 3, 4, 2, 3, 4, 5, 3, 4, 2, 2, 3, 7, 6, 5, 6, 3, 4, 5, 4, 9, 4, 6, 6, 3, 7, 7, 5, 5, 4, 5

Offset: 1

Zhi-Wei Sun, Jan 06 2009

nonn

Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a cube of a positive Fibonacci number, with one of two Fibonacci numbers odd. He has verified this up to 3*10^7.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

			For n=14 the a(14)=4 solutions are 3+F_4+(F_3)^3, 5+F_2+(F_3)^3, 5+F_6+(F_2)^3, 11+F_3+(F_2)^3

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

Zhi-Wei SUN, Table of n, a(n), n=1..50000.
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
K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp., in press. arXiv:math.NT/0702382

Cf. A154257, A154258, A000040, A000045

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n,2]==0||Mod[x,3]>0)&&PQ[n-(Fibonacci[x])^3-Fibonacci[y]],1,0], {x,2,2*Log[2,n^(1/3)+1]},{y,2,2*Log[2,Max[2,n-(Fibonacci[x])^3]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]

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A154285 Number of ordered triples satisfying p+L_s+L_t=n, where p is an odd prime, s and t are nonnegative and the Lucas number L_s or L_t is odd.

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A155114 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.

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A154263 Number of triples such that p+F_s+(F_t)^3=n, where p is an odd prime, s and t are greater than one and F_s or F_t is odd.

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