A256558 Number of ways to write n = p + floor(k*(k+1)/4), where p is a prime and k is a positive integer.

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

Offset: 1

Author

Zhi-Wei Sun, Apr 01 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1.

(ii) For any integer m > 4 not equal to 12, each integer n > 1 can be written as p + floor((k^2-1)/m), where p is a prime and k is a positive integer.

We also have some other conjectures on representations n = p + floor(k*(k+1)/m) with m > 4.

Examples

			 a(15) = 1 since 15 = 5 + floor(6*7/4) with 5 prime.
a(420) = 1 since 420 = 419 + floor(2*3/4) with 419 prime.
a(945) = 1 since 945 = 877 + floor(16*17/4) with 877 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.
• Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.

Crossrefs

Cf. A000040, A000217, A132399, A256544, A256555.

Programs

Mathematica

Do[r=0;Do[If[PrimeQ[n-Floor[k(k+1)/4]],r=r+1],{k,1,(Sqrt[16n+1]-1)/2}];Print[n," ",r];Continue,{n,1,100}]