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Template:Sequence of the Day for February 5

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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 + 2x + 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 + 2x + 2  y
with
p
an odd prime and
x
,
y
positive integers.