Template:Sequence of the Day for January 29

Intended for: January 29, 2014

Timetable

• First draft entered by Alonso del Arte on January 18, 2013 ✓
• Draft reviewed by Daniel Forgues on January 30, 2016 ✓, January 28, 2020 ✓
• Draft to be approved by December 29, 2013

Yesterday's SOTD * Tomorrow's SOTD

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A082813: Prime numbers of the form

∑ ω (m) i  = 1 αi  pi m

for some composite number

m = ∏ ω (m) i  = 1 pi αi

.

{2 ⋅  2 2 2 ⋅  3 + 3 2 2 ⋅  3, 3 ⋅  2 2 3 ⋅  11 + 11 2 3 ⋅  11, 2 ⋅  3 3 2 ⋅  23 + 23 3 2 ⋅  23, ...} = {539633, ... }

Already, the second term has 92 decimal digits and the third term has 282 decimal digits. There are potentially a few more terms that are not known (it is not known whether the sequence is infinite or not).

Although getting terms of this sequence is computationally hard, it is at least possible to quickly cull out many

m

that would simply not work, such as

m

of the form

2 3 3 α

(because then the resulting number, though quite large, would be divisible by 3). ^{(Provide proof: PROOF GOES HERE. □)}