Template:Sequence of the Day for March 10

Intended for: March 10, 2015

Timetable

• First draft entered by M. F. Hasler on March 9, 2014 ✓
• Draft reviewed by Alonso del Arte on March 10, 2014 ✓
• Draft reviewed by Daniel Forgues on March 10, 2016 ✓
• Draft to be approved by February 10, 2015

Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.

---

A132995

a (n) = gcd (  ∑ n k  = 1  p (k )  , ∏ n   j  = 1  p (  j ) )

, where

p (k )

is the

k

-th prime.

{ 2, 1, 10, 1, 14, 1, 2, 77, 10, 3, 10, ..., 2014, 3, 14, ... }

That is, A132995 (n) = GCD (A007504 (n), A002110 (n)).

It may be astonishing that the GCD of sum (A007504) and product (A002110) of the first

n

primes displays such irregular behavior. But some of the patterns find easy explanations:

• Every second term (n = 2, 4, 6, ...) of the sum A007504 (n) is odd, so

  a (2 k )

  cannot be even. It turns out that

  a (2 k )

  is often (but not always) equal to 1, for small k = 1, ..., 7, then often equal to 3, for k = 5, ..., 23.
• Similarly, A007504 (n) is even for odd n = 2 k  −  1 = 1, 3, 5, 7, ..., therefore

  a (2 k  −  1)

  is also always even.
• Taking the GCD with the product A002110 (n) amounts to have

  a (n)

  equal to the product of all those among the first

  n

  primes that divide the sum A007504 (n).
• For larger terms, it is increasingly probable that A007504 (n) has several of the smaller primes as factors. But looking at the very interesting graph of the sequence, it seems that there are infinitely many

  a (n) = 1

  ,

  a (n) = 2

  and

  a (n) = 3

  .

Trivia

This is the only sequence which has the terms “3, 10” and later “2014,” within the displayed 3 lines of data. (It would have been a nice choice for this year’s March 14, too, since the 2014 is followed by 3, 14! But there is already another Sequence of the Day for March 14 related to

π = 3.14...

)

A search for “3, 10, ..., 2012” yields A212068 and “3, 10, ..., 2015” yields A202339 as only result, while no result is found for 2010, 2011, 2013.