A338274 Number of decompositions of 2*n >= 4 into an unordered sum of two primes, n - d and n + d, such that d < sqrt(2*n - 1).

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 2, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 1, 0, 2, 2, 1, 2, 1, 0, 1, 2, 0, 1, 2, 1, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 2, 2, 0, 2, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 1, 0, 1

Offset

2,4

Comments

Indices of the terms with a(n) = 0 are the 1225 numbers in sequence A336585, in which the last term is 1353559. Conjecture: any even number 2*n with n > 1353559 can be written as the sum of two primes, p = m - d and q = m + d, such that d < sqrt(2*m - 1). Note that, for the Goldbach conjecture, the half gap d between the two prime numbers is (q - p)/2 <= m - 2. For the stronger Goldbach conjecture made in A335225, d < m/2. Thus, the conjecture made here is a more stronger version of the Goldbach conjecture.

The record values of a(n) and corresponding n (in parentheses) for a(n) up to 140 are: 1(2), 2(5), 3(105), 4(165), 5(585), 6(630), 7(1575), 8(2685), 10(2730), 11(6870), 12(11970), 13(12390), 14(21540), 15(23925), 16(32280), 17(41745), 19(42315), 20(55965), 21(59340), 22(89985), 23(99330), 24(124950), 25(145740), 26(150150), 27(185955), 30(192045), 31(207900), 34(303765), 35(392595), 38(431655), 46(464100), 47(879060), 50(1080450), 51(1128435), 53(1322685), 55(1340955), 58(1477875), 60(1601985), 61(1909215), 65(1996995), 66(2486715), 68(2513280), 70(2656500), 71(2917530), 74(3366825), 75(3566640), 77(3610530), 79(4037880), 82(4160520), 85(4834830), 90(4868955), 96(6006000), 98(7843290), 101(8303295), 102(8918910), 108(9503340), 113(9725100), 116(11539605), 119(11800635), 120(13311480), 133(13783770), 137(16956225), 140(17402385). Conjecture: values of n at which record values of a(n) (>=2) are achieved are multiples of 5.

Links

Table of n, a(n) for n=2..88.

Example

a(2) = 1 because 2*n = 2 + 2 and d = 0 < sqrt(2*2 - 1);
a(5) = 2 because 2*n = 5 + 5 = 3 + 7 and both d's ( 0 and 2) < sqrt(2*5 - 1);
a(22) = 0 because none of the values of d (9, 15 and 19) for the three Goldbach pairs of 2*22 (13&31, 7&37 and 3&41) is < sqrt(2*22 - 1);
a(105) = 3 because the values of d (2, 4, and 8) for 3 of the 19 Goldbach pairs are < sqrt(2*105 - 1).

Prog

(Python)
from sympy import isprime
from math import sqrt
m = 2
while m >= 2:
    d = 0
    a = 0
    while d < sqrt(2*m - 1):
        p = m - d
        q = m + d
        if isprime(p) == 1 and isprime(q) == 1:
            a += 1
        d += 1
    print (a)
    m += 1

Crossrefs

Cf. A002375, A335225, A336585.

Sequence in context: A154843 A062557 A276438 * A210960 A339095 A193509

Adjacent sequences: A338271 A338272 A338273 * A338275 A338276 A338277

Keyword

nonn

Author

Ya-Ping Lu, Oct 19 2020

Status

approved