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.

A190006 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sin(Pi/3), t=csc(Pi/3).

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 21, 23, 26, 29, 32, 35, 39, 42, 44, 47, 50, 53, 56, 60, 63, 66, 68, 71, 74, 78, 81, 84, 87, 89, 92, 95, 99, 102, 105, 108, 111, 113, 117, 120, 123, 126, 129, 132, 134, 138, 141, 144, 147, 150, 153, 157, 159, 162, 165, 168, 171, 174, 178, 180, 183, 186, 189, 192, 196, 199, 202, 204, 207, 210, 213, 217, 220
Offset: 1

Views

Author

Clark Kimberling, May 03 2011

Keywords

Comments

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n) = n + [n*s/r] + [n*t/r],
b(n) = n + [n*r/s] + [n*t/s],
c(n) = n + [n*r/t] + [n*s/t], where []=floor.
Taking r=1, s=sin(Pi/3), t=csc(Pi/3) gives a=A190006, b=A190007, c=A190008.

Links

Crossrefs

Cf. A180007, A190008.

Programs

  • Magma
    C := ComplexField(); [n + Floor(n*Sin(Pi(C)/3)) + Floor(n/Sin(Pi(C)/3)): n in [1..100]]; // G. C. Greubel, Jan 11 2018
  • Mathematica
    r=1; s=Sin[Pi/3]; t=Csc[Pi/3];
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (* A190006 *)
Table[b[n], {n, 1, 120}]  (* A190007 *)
Table[c[n], {n, 1, 120}]  (* A190008 *)
  • PARI
    for(n=1,100, print1(n + floor(n*sin(Pi/3)) + floor(n/sin(Pi/3)), ", ")) \\ G. C. Greubel, Jan 11 2018

Formula

A190006: a(n) = n + [n*sin(Pi/3)] + [n*csc(Pi/3)].
A190007: b(n) = n + [n*csc(Pi/3)] + [n*(csc(Pi/3))^2].
A190008: c(n) = n + [n*sin(Pi/3)] + [n*(sin(Pi/3))^2].

A180041 Number of Goldbach partitions of (2n)^n.

Original entry on oeis.org

0, 2, 13, 53, 810, 20564, 274904, 6341424, 419586990
Offset: 1

Views

Author

Jonathan Vos Post, Aug 07 2010

Keywords

Comments

This is the main diagonal of the array mentioned in A180007, only considering even rows (as odd numbers cannot be the sums of two odd primes), namely A(2n, n) = number of ways of writing (2n)^n as the sum of two odd primes, when the order does not matter.

Examples

			a(1) = 0 because 2*1 = 2 is too small to be the sum of two primes.
a(2) = 2 because 4^2 = 16 = 3+13 = 5+11.
a(3) = 13 because 6^3 = 216 and A180007(3) = Number of Goldbach partitions of 6^3 = 13.
a(4) = 53 because 8^4 = 2^12 and A006307(12) = Number of ways writing 2^12 as unordered sums of 2 primes.

Crossrefs

Cf. A001031, A061358, A065577, A180007.

Programs

  • Maple
    A180041 := proc(n) local a,m,p: if(n=1)then return 0:fi: a:=0: m:=(2*n)^n: p:=prevprime(ceil((m-1)/2)): while p > 2 do if isprime(m-p) then a:=a+1: fi: p := prevprime(p): od: return a: end: seq(A180041(n),n=1..5); # Nathaniel Johnston, May 08 2011
  • Mathematica
    f[n_] := Block[{c = 0, p = 3, m = (2 n)^n}, lmt = Floor[m/2] + 1; While[p < lmt, If[ PrimeQ[m - p], c++ ]; p = NextPrime@p]; c]; Do[ Print[{n, f@n // Timing}], {n, 8}] (* Robert G. Wilson v, Aug 10 2010 *)

Formula

a(n) = A061358((2*n)^n) = A061358(A062971(n)).

Extensions

a(6)-a(8) from Robert G. Wilson v, Aug 10 2010
a(9) from Giovanni Resta, Apr 15 2019

A195295 Number of Goldbach partitions of 4^n.

Original entry on oeis.org

0, 1, 2, 5, 8, 22, 53, 151, 435, 1314, 4239, 13705, 45746, 153850, 525236, 1817111, 6341424, 22336060, 79287664, 283277225, 1018369893
Offset: 0

Views

Author

Kausthub Gudipati, Sep 16 2011

Keywords

Comments

Bisection of A006307.

Crossrefs

Cf. A061358, A065577, A180007.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.