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.

A178129 Partial sums of A050508.

Original entry on oeis.org

0, 2, 8, 23, 47, 87, 147, 224, 328, 463, 623, 821, 1049, 1322, 1644, 2004, 2420, 2896, 3418, 4007, 4647, 5361, 6153, 7004, 7940, 8940, 10032, 11220, 12480, 13843, 15313, 16863, 18527, 20276, 22146, 24141, 26229, 28449, 30767, 33224, 35824, 38530
Offset: 0

Views

Author

Jonathan Vos Post, May 20 2010

Keywords

Comments

Partial sums of golden rectangle numbers. The subsequence

Examples

			a(19) = 0 + 2 + 6 + 15 + 24 + 40 + 60 + 77 + 104 + 135 +

Crossrefs

Cf. A000201, A007064, A026355, A007067, A050508.

Programs

  • Python
    from math import isqrt
from itertools import count, islice
def A178129_gen(): # generator of terms
    return accumulate(n*((isqrt(5*n**2<<2)>>1)+n+1>>1) for n in count(0))
A178129_list = list(islice(A178129_gen(),10)) # Chai Wah Wu, Aug 29 2022

Formula

a(n) = Sum_{i=0..n} A050508(i) = Sum_{i=0..n} (i*A007067(i)).

A108540 Golden semiprimes: a(n)=p*q and abs(p*phi-q)<1, where phi = golden ratio = (1+sqrt(5))/2.

Original entry on oeis.org

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2005; revised Jun 13 2005

Keywords

Examples

			589 = 19*31 and abs(19*phi - 31) = abs(30,7426... - 31) < 1, therefore 589 is a term.

Links

Crossrefs

Cf. A001622, A001358, A050508.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; q = f[p]; If[q > 0, AppendTo[seq, p*q]], {100}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A108541(n)*A108542(n) = A000040(k)*A108539(k) for some k.

Extensions

Corrected by T. D. Noe, Oct 25 2006

A050510 Golden rectangular box numbers: a(n) = n*A007067(n)*A007067(A007067(n)).

Original entry on oeis.org

0, 6, 30, 120, 240, 520, 960, 1386, 2184, 3240, 4160, 5742, 7068, 9282, 11914, 14040, 17472, 21420, 24534, 29450, 33280, 39270, 45936, 51060, 58968, 65000, 74256, 84348, 91980, 103588, 116130, 125550, 139776, 150414, 166430, 183540, 196272
Offset: 0

Views

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 28 1999

Keywords

Crossrefs

Cf. A007067, A050508.

Programs

  • Mathematica
    b[n_] := Round[n*GoldenRatio];
a[n_] := n*b[n]*b[b[n]];
Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Dec 15 2017 *)
  • PARI
    b(n) = round(n*(1+sqrt(5))/2);
a(n) = n*b(n)*b(b(n)); \\ Michel Marcus, Dec 15 2017

Formula

a(n) = n*round(n*(1+sqrt(5))/2)*round(round(n*(1+sqrt(5))/2)*(1+sqrt(5))/2). - Wesley Ivan Hurt, Apr 23 2021
Showing 1-3 of 3 results.

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

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