Template:Sequence of the Day for February 5

Intended for: February 5, 2013

Timetable

• First draft entered by Alonso del Arte on November 3, 2011 based on remarks from Zhi Wei-Sun ✓
• Draft reviewed by Alonso del Arte on December 1, 2012 ✓
• Draft to be approved by January 5, 2012

Yesterday's SOTD * Tomorrow's SOTD

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A157218: 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 1 mod 6 and

x

,

y

positive integers.

{ ... 0, 1, 1, 0, 2, 1, 0, 2, 3, 1, ... }

In 2009, Zhi-Wei Sun conjectured that

a (n) > 0

for all

n > 17

; in other words, any odd integer greater than 34 can be written as the sum of a prime congruent to 1 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 R. 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.