cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A079098 Conjectured values of greatest k such that for any consecutive primes q, q', k <= q < q', sqrt(q')-sqrt(q) < 1/n.

Original entry on oeis.org

1, 113, 1327, 2971, 31397, 34061, 43331, 44293, 58831, 155921, 370261, 370261, 492113, 492113, 492113, 604073, 604073, 1357201, 1561919, 2010733, 2010733, 2010733, 2010733, 2010733, 2010733, 2010733, 2010733, 2238823, 4652353, 4652353, 4652353, 4652353
Offset: 1

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Author

Rainer Rosenthal, Feb 02 2003

Keywords

Comments

Inspired by Andrica's conjecture.
Each of these terms, k, has been tested to at least 100*k. - Sean A. Irvine, Jul 29 2025

References

  • R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21

Links

Crossrefs

Cf. A038458, A074976, A078693.

Extensions

More terms from Sean A. Irvine, Jul 29 2025

A252477 Integer part of 1/(sqrt(prime(n+1))-sqrt(prime(n))).

Original entry on oeis.org

3, 1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 3, 6, 3, 2, 2, 7, 2, 4, 8, 2, 4, 3, 2, 4, 10, 5, 10, 5, 1, 5, 3, 11, 2, 12, 4, 4, 6, 4, 4, 13, 2, 13, 6, 14, 2, 2, 7, 15, 7, 5, 15, 3, 5, 5, 5, 16, 5, 8, 16, 3, 2, 8, 17, 8, 2, 6, 3, 18, 9, 6, 4, 6, 6, 9, 6, 4, 9, 5, 4, 20, 4, 20, 6, 10, 7, 5, 10, 21, 10, 3, 5
Offset: 1

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Author

M. F. Hasler, Dec 31 2014

Keywords

Comments

Andrica's conjecture states that sqrt(prime(n+1))-sqrt(prime(n)) < 1 for all n. Since equality cannot happen, this is equivalent to say that all terms of is sequence are >= 1.
Sequence A074976 is based on the same idea (rounding to the nearest integer instead).
It is a remarkable coincidence(?) that very often, especially around "peaks", a symmetric pattern "x, y, x" occurs: 2, 7, 2,... 10, 5, 10,... 13, 2, 13,... 20, 4, 20, ..., 11, 5, 11, ...
Equal to the integer part of (A000006(n+1)+A000006(n))/(prime(n+1)-prime(n)) for most indices; exceptions are 1, 129, 1667, 2004, 2088, 2334, 3377, 3585, 3695, 3834, 4978, 7057, 7950, 8103, 9525, 9805,...

Examples

			a(1) = floor(1/(sqrt(3) - sqrt(2))) = floor(1/(1.73-1.41)) = floor(1/0.32) = floor(3.15) = 3.
a(2) = floor(1/(sqrt(5) - sqrt(3))) = floor(1/(2.236-1.732)) = floor(1/0.504) = floor(1.98) = 1.

Links

Crossrefs

Cf. A000040, A074976, A079636.

Programs

  • Haskell
    a252477 n = a252477_list !! (n-1)
a252477_list = map (floor . recip) $ zipWith (-) (tail rs) rs
               where rs = map (sqrt . fromIntegral) a000040_list
-- Reinhard Zumkeller, Jan 04 2015
  • Maple
    a:= n-> ((w, v)-> floor(1/(w-v)))(map(sqrt@ithprime, [n+1, n])[]):
seq(a(n), n=1..92);  # Alois P. Heinz, Aug 23 2025
  • Mathematica
    Floor[1/Subtract @@@ Reverse[Partition[Sqrt[Prime[Range[100]]], 2, 1], 2]] (* Paolo Xausa, Aug 24 2025 *)
  • PARI
    a(n)=1\(sqrt(prime(n+1))-sqrt(prime(n))) \\ M. F. Hasler, Dec 31 2014

Formula

a(n) = A079636(n) - 1. - Alois P. Heinz, Aug 23 2025

A079063 Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1.

Original entry on oeis.org

3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 5, 4, 4, 4, 4, 5, 6, 5, 4, 4, 3, 3, 5, 5, 5, 5, 6, 5, 6, 5, 6, 7, 6, 5, 5, 4, 4, 4, 7, 7, 7, 6, 6, 6, 6, 8, 7, 7, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 7, 8, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 8, 8, 9, 9, 8, 8, 7, 8, 8, 8, 7, 7, 8, 7, 6, 6, 6, 5, 6, 6, 8, 8, 9, 9, 10
Offset: 1

Views

Author

Benoit Cloitre, Feb 02 2003

Keywords

Comments

Inspired by Andrica's conjecture. If it is true, a(n)>1 for all n.
Cf. A038458, A074976, A078693

Links

Programs

  • PARI
    a(n)=if(n<0,0,k=1; while(abs(sqrt(prime(n+k))-sqrt(prime(n)))<1,k++); k)

Formula

Conjecture: there is a constant c>0 such that for n large enough, a(n)>c*sqrt(n) and we can take c=0.4. More precisely, there are 2 constants A and B such that A=lim sup n ->infinity a(n)/sqrt(n) exists = 0.75....; B=lim inf n ->infinity a(n)/sqrt(n) exists =0.46....
Showing 1-3 of 3 results.

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