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A256558
Number of ways to write n = p + floor(k*(k+1)/4), where p is a prime and k is a positive integer.
2
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
1,3
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.
LINKS
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.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 01 2015
STATUS
approved