#oeis

i’ll block a bitch on the OEIS

Happy new year! Hypergraphs are a generalization of graphs; trying to count hyperforests without seeds (i.e. without isolated nodes) leads to this integer sequence, containing 2026…

the Online Encyclopedia of Integer Sequences is failing to load so basically the world is over

A fun thing I found on the OEIS (online encyclopedia of integer sequences):

Suppose you have a sequence like

1,2,2,3,3,3,4,4,4,4…

The pattern is “each number n appears n times”. You can make this sequence in a very unexpected way with squareroots.

If you round sqrt(2x) to the nearest whole numbers, you also get the same sequence!

Interesting math fact of the day #176:

There are only 2 (surprisingly simple) OEIS sequences with the subarray “7, 3, 73”.

A379641 - OEIS

A381493 - OEIS

Interesting math fact of the day #102:

2000020^64 + 1 is prime

The OEIS is really weird sometimes

A156085 - OEIS

Ghee Beom Kim Double Spiral

Stuart Errol Anderson commented:

A090390 - OEIS

1, 1, 9, 49, 289, 1681, 9801, 57121, 332929, 1940449, 11309769, 65918161, 384199201, 2239277041, 13051463049, 76069501249, 443365544449, 2584123765441, 15061377048201, 87784138523761, 511643454094369, 2982076586042449, 17380816062160329, 101302819786919521, 590436102659356801

Square of the Pell-Lucas numbers, “numerators of continued fraction convergents to sqrt(2).”

The ratios converge (of course?)

OEIS has ‘The ratio a(n+1)/a(n) converges to 3 + 2*sqrt(2). - Richard R. Forberg, Aug 14 2013’ and 'a(n) = (((1+sqrt(2))^(2n) + (1-sqrt(2))^(2n)) + 2*(-1)^n)/4 - Lambert Klasen’

And someone found Rick Mawbry’s page for this related double spiral…

Which must mean that A090390 must also give a sum for ½…

1/(2+1/r)(1+1/r+1/r^2+…)=½ where r is the limit ratio.

Oh! That doesn’t work because the perfect ratio doesn’t make a rectangle!

Of course I had to make in GeoGebra.

i will never cease to be amused by the oeis website

Before the discovery of the OEIS, there were actually people called “sequencers” who memorized every sequence, and were trained in analytical skills to be able to find sequences given subsequences. 

So I noticed that today, 12/22/21 is a palindrome date, and also if written as on a calculator an upside-down palindrome date, and also 122221 = 11 * 11111, so this is the last date that is in integer sequence A308365 (numbers which are products of repunits) for a long time. And I thought that was neat.

A great introduction to the Online Encyclopedia of Integer Sequences by means of Adele’s studio albums.

^pipis

On-line Encyclopedia of Integer Sequences

A006880 Number of primes < 10^n

0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290 

• There are 0 primes below 10^0 (1)
• There are 4 primes below 10^1 (10)
• There are 25 primes below 10^2 (100)
• etc.

This shows that primes become less frequent as you go on

• 40% of numbers under 10 are prime
• 25% of numbers under 100 are prime
• 16.8% under 1,000
• 12.29% under 10,000
• 9.592% under 100,000
• 7.8498% under 1,000,000
• etc.

3, 5, and 7 are the only numbers below which a majority are prime, though 2, 4, 6, and 8 come close.

1. not prime (0/1, 0%)
   
2. prime (½, 50%)
   
3. prime (2/3, 66.6…%, highest percentage of primes)
   
4. not prime (2/4, 50%, most common percentage of primes)
   
5. prime (3/5, 60%)
   
6. not prime (3/6, 50%)
   
7. prime (4/7, 57.142857…%)
   
8. not prime (4/8, 50%)
   
9. not prime (4/9, 44.4…%)
   
10. not prime (4/10, 40%)
11. prime (5/11, 45.45…%)
12. not prime (5/12, 41.6…%)
13. prime (6/13, 46.153846…%)
14. not prime (6/14,  42.857142…%)
15. not prime (6/15,  40%)
16. not prime (6/15, 37.5%)
    
17. prime (7/17, 41.17647059…%)
    
18. not prime (7/18, 38.8…%)
    
19. prime (8/19, 42.10526316…%)
    
20. not prime (8/20, 40%)
    
21. not prime (8/21, 38.0952381…%)
    
22. not prime (8/22, 36.36…%)
    
23. prime (9/23, 39.13043478…%)
    
24. not prime (9/24, 37.5%)
    
25. not prime (9/25, 36%)
    
26. not prime (9/26, 34.61538462…%)
    
27. not prime (9/27, 33.3…%)
    
28. not prime (9/28, 32.142857…%)
    
29. prime (10/29, 34.4827586207%)
    
30. not prime (10/30, 33.3…%)

This post has no larger point, I just wanted to show this off

Unfortunately, there is no Online Encyclopedia of Real Sequences like there is the OEIS

But we could always just construct an encyclopedia of rational sequences out of the integer sequences, and from there construct the real sequences as equivalence classes of cauchy sequences OF rational sequences…..

evil CRCs can detect any odd amounts of bit errors

You know who you are.

@the-real-numbers

Duuust in the wind… all we 

“[the] sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression” 

are is, is duuuuust in the wind… 

I think the Online Encyclopedia of Integer Sequences is my best friend.

Ma il gas sarà azero?

O ci sarà (a)zero gas?

Ci apprestiamo ad espiantare gli olivi salentini per un itinerario del costruendo gasdotto totalmente ingiustificato e vengono già alla luce loschi intrecci di interessi locali.

Ma nessuno si chiede se sarà azero il gas destinato all’Italia.

Dicono che il gasdotto servirà a diversificare le fonti di approvvigionamento, affrancandoci dai russi, ma ci sono buone probabilità…

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