A154285 -id:A154285 - cp's OEIS frontend

A154421 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.

0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 2, 5, 2, 4, 3, 4, 4, 5, 2, 6, 2, 7, 5, 7, 3, 9, 3, 9, 4, 7, 3, 6, 4, 9, 3, 10, 3, 8, 4, 6, 5, 8, 6, 8, 3, 9, 4, 8, 6, 8

Offset: 1

Zhi-Wei Sun, Jan 09 2009

nonn

On Jan 09 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17 2009, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.

			For n=8 the a(8)=3 solutions are 3 + F_4 + L_0, 3 + F_2 + L_3, 5 + F_2 + L_0.

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), 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)
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, [DOI], arXiv:math.NT/0702382

Cf. A000040, A000045, A000032, A154257, A154285, A154290, A154364, A154417, A156695.

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

a(n) = |{: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|.

McNeil's counterexample added by Zhi-Wei Sun, Jan 20 2009

A157225 Number of ways to write the n-th positive odd integer in the form p+2^x+7*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 3, 1, 2, 4, 1, 2, 4, 2, 2, 3, 2, 2, 4, 2, 4, 4, 1, 5, 5, 2, 5, 7, 1, 3, 7, 2, 4, 8, 2, 4, 3, 2, 4

Offset: 1

Zhi-Wei Sun, Feb 25 2009

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<11 or n=13,16,992; in other words, except for 25, 31, 1983, any odd integer greater than 20 can be written as the sum of a prime congruent to 5 mod 6, a positive power of 2 and seven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

			For n=18 the a(18)=3 solutions are 2*18-1=5+2+7*2^2=5+2^4+7*2=17+2^2+7*2.

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) for n=1..200000
Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

Cf. A000040, A000079, A157218, A155860, A155904, A156695, A154257, A154285, A155114, A154536, A154404, A154940.

PQ[x_]:=x>1&&Mod[x,6]==5&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-7*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/7]},{y,1,Log[2,Max[2,2n-1-7*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

a(n)=|{: p+2^x+7*2^y=2n-1 with p a prime congruent to 5 mod 6 and x,y positive integers}|

A157242 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 25 2009

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<15 or n=17, 20, 23, 86, 124; in other words, except for 33, 39, 45, 171 and 247, any odd integer greater than 28 can be written as the sum of a prime p=5 (mod 6), a positive power of 2 and eleven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7. Knowing the conjecture from Sun, Qing-Hu Hou and D. S. McNeil have continued the verification for odd integers below 1.5*10^8 and 10^12 respectively, and they have found no counterexample. Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

			For n=18 the a(18)=2 solutions are 2*18-1=5+2^3+2*11=11+2+2*11.

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) for n=1..200000
Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

A000040, A000079, A157237, A155860, A155904, A156695, A154257, A154285, A155114, A154536

PQ[x_]:=x>1&&Mod[x,6]==5&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-11*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/11]},{y,1,Log[2,Max[2,2n-1-11*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

a(n)=|{: p+2^x+11*2^y=2n-1 with p a prime congruent to 5 mod 6 and x,y positive integers}|

A157372 Number of ways to write the (n+50)-th positive odd integer in the form p+2^x+51*2^y with p an odd prime and x,y positive integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 28 2009

Zhi-Wei Sun guessed that a(n)=0 if and only if n=1,2,3,127; in other words, except for 353, any odd integer greater than 105 can be written as the sum of an odd prime, a positive power of two and 51 times a positive power of two. D. S. McNeil has verified this for odd integers below 10^12. This is a part of the project for the form p+2^x+k*2^y with k=3,5,...,61 initiated by Zhi-Wei Sun in Jan. 2009.

			For n = 9 the a(9) = 3 solutions are: 2*59-1 = 7+2^3+51*2 = 11+2^2+51*2 = 13+2+51*2.

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) for n=1..200000
Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A curious conjecture about p+2^x+11*2^y
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

Cf. A000040, A000079, A157237, A157242, A155860, A155904, A156695, A154257, A154285, A155114, A154536.

PQ[x_]:=x>1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2(n+50)-1-51*2^x-2^y],1,0], {x,1,Log[2,(2(n+50)-1)/51]},{y,1,Log[2,Max[2,2(n+50)-1-51*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

a(n)=|{: p+2^x+51*2^y=2(n+50)-1 with p an odd prime and x,y positive integers}|

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A154421 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.

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A157225 Number of ways to write the n-th positive odd integer in the form p+2^x+7*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

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A157242 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

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A157372 Number of ways to write the (n+50)-th positive odd integer in the form p+2^x+51*2^y with p an odd prime and x,y positive integers.

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