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 1-4 of 4 results.

A006719 a(n) = A000905(n) + 1.

Original entry on oeis.org

3, 4, 6, 12, 48, 924, 409620, 83763206256, 3508125906290858798172, 6153473687096578758448522809275077520433168, 18932619208894981833333582059033329370801266249535902023330546944758507753065602135844
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Related to Hamilton numbers.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • J. J. Sylvester, Collected Mathematical Papers, Vols. 1-4, Cambridge Univ. Press, 1904-1912, Vol. 4, p. 551.

Links

  • J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.

Extensions

More terms from Ralf Stephan, Aug 05 2004

A001660 Hypotenusal numbers.

Original entry on oeis.org

1, 1, 2, 6, 36, 876, 408696, 83762796636, 3508125906207095591916, 6153473687096578758445014683368786661634996, 18932619208894981833333582059033329370801260096062214926751788496235698477988081702676
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • J. J. Sylvester and M. J. Hammond, On Hamilton's numbers, Phil. Trans. Roy. Soc., 178 (1887), 285-312.

Links

Crossrefs

First differences of A000905.

Programs

  • Mathematica
    h[1] = 2; h[n_] := h[n] = 2+Sum[(-1)^(i+1)*Product[h[n-i]-k, {k, 0, i}]/(i+1)!, {i, 1, n-1}]; a[0] = 1; a[n_] := h[n+1] - h[n]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 05 2013 *)

A134294 "Maximal" Hamilton numbers. Differs from usual Hamilton numbers starting at n=4.

Original entry on oeis.org

2, 3, 5, 10, 44, 906, 409181, 83762797734
Offset: 1

Views

Author

Olivier Gérard, Oct 17 2007

Keywords

Comments

a(n) is the minimal degree of an equation from which n successive terms after the first can be removed (by a series of transformation comparable to Tschirnhaus's) without requiring the solution of at least one irreducible equation of degree greater than n. The cases where an equation of degree greater than n is needed but is in fact factorizable into several equations of degree all less than or equal to n are considered as fair. a(n) <= A000905(n) by definition.

Examples

			a(4)=10 because one can remove 4 terms in an equation of degree 10 by solving two quartic equations.

References

  • W. R. Hamilton, Sixth Report of the British Association for the Advancement of Science, London, 1831, 295-348.

Links

Crossrefs

Cf. A000905.

A217864 Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).

Original entry on oeis.org

0, 2, 2, 2, 0, 2, 1, 2, 2, 1, 1, 2, 0, 1, 2, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1
Offset: 1

Views

Author

Jon Perry, Oct 13 2012

Keywords

Comments

Conjecture: a(n) is unbounded.
If Riemann Hypothesis is true, this is probably true as the PNT is generally a lower bound for Pi(n).
Conjecture: a(n)=0 infinitely often.
The first conjecture follows from Dickson's conjecture. The second conjecture follows from a theorem of Brauer & Zeitz on prime gaps. - Charles R Greathouse IV, Oct 15 2012

Examples

			log(1)=0 and 2*log(2) ~ 1.38629436112. Hence, a(1)=0.
Floor(2*log(2)) = 1 and 3*log(3) ~ 3.295836866. Hence, a(2)=2.

References

  • A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.

Links

Crossrefs

An alternate version of A166712.
Cf. A217564, A096509, A000905, A050504, A000720.

Programs

  • JavaScript
    function isprime(i) {
if (i==1) return false;
if (i==2) return true;
if (i%2==0) return false;
for (j=3;j<=Math.floor(Math.sqrt(i));j+=2)
if (i%j==0) return false;
return true;
}
for (i=1;i<88;i++) {
c=0;
for (k=Math.floor(i*Math.log(i));k<=(i+1)*Math.log(i+1);k++) if (isprime(k)) c++;
document.write(c+", ");
}
  • Mathematica
    Table[s = Floor[n*Log[n]]; PrimePi[(n+1) Log[n+1]] - PrimePi[s] + Boole[PrimeQ[s]], {n, 100}] (* T. D. Noe, Oct 15 2012 *)
  • PARI
    a(n)=sum(k=n*log(n)\1,(n+1)*log(n+1),isprime(k)) \\ Charles R Greathouse IV, Oct 15 2012
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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