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.

User: David Garber

David Garber's wiki page.

David Garber has authored 13 sequences. Here are the ten most recent ones:

A132475 Numerical equivalents for the 23 Latin letters, according to Tartaglia.

Original entry on oeis.org

500, 300, 100, 500, 250, 40, 400, 200, 1, 51, 50, 1000, 90, 11, 400, 500, 80, 70, 160, 5, 10, 150, 2000
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2007, based on email from David Garber and on material from the Roman Numerals web site mentioned above, especially a posting by Peter T. Daniels. Thanks to M. F. Hasler for finding this web site

Keywords

Examples

			A 500
B 300
C 100
D 500
E 250
F 40
G 400
H 200
I 1
K 51
L 50
M 1000
N 90
O 11
P 400
Q 500
R 80
S 70
T 160
V 5
X 10
Y 150
Z 2000

References

  • A. R. Bradford, Crossword Lists, Collins.
  • Florian Cajori, A History of Mathematical Notations (1928, repr. Dover, 1993), para. 60.
  • G. Friedlein, Die Zahlzeichen und das elementare Rechnen der Griechen und Roemer, Erlangen, 1869.
  • Tartaglia, General Trattato di Numeri, Part I (1556), folios 4, 5.

Links

A113943 Primes arising in A073946.

Original entry on oeis.org

13, 47, 103, 151, 433, 739, 1123, 3581, 7591, 9319, 10789, 11903, 14543, 14797, 22129, 28069, 42043, 43331, 44203, 47779, 49613, 50539, 52903, 66617, 70423, 87313, 87931, 88547, 91673, 94841, 98713, 102677, 113591, 128747, 133261
Offset: 1

Views

Author

David Garber, Nov 13 2002

Keywords

A113944 Square roots of A073946.

Original entry on oeis.org

3, 6, 9, 11, 19, 25, 31, 56, 82, 91, 98, 103, 114, 115, 141, 159, 195, 198, 200, 208, 212, 214, 219, 246, 253, 282, 283, 284, 289, 294, 300, 306, 322, 343, 349, 360, 373, 378, 384, 393, 414, 427, 439, 440, 454, 459, 461, 464, 473, 474, 491, 521
Offset: 1

Views

Author

David Garber, Nov 13 2002

Keywords

A110887 Number of configurations of n skew lines up to isotopy.

Original entry on oeis.org

1, 1, 2, 3, 7, 19, 74
Offset: 1

Views

Author

David Garber, Sep 20 2005

Keywords

References

  • A. Borobia and V. F. Mazurovskii, On diagrams of configurations of 7 skew lines in R^3, Amer. Math. Soc. Transl. (2) 173 (1996), 33-40
  • A. Borobia and V. F. Mazurovskii, Nonsingular configurations of 7 lines in RP^3, J. Knot Theory Ramif. 6(6) (1997), 751-783
  • O. Y. Viro and Y. V. Drobotukhina, Configurations of skew lines, Leningrad Math. J. 1(4) (1990), 1027-1050

A110888 Number of spindle configurations of n skew lines.

Original entry on oeis.org

1, 1, 2, 3, 7, 15, 48, 180, 985, 6867, 60108, 609112, 6909017
Offset: 1

Views

Author

David Garber, Sep 20 2005

Keywords

Links

A110890 Number of spindle configurations of n skew lines, up to isotopy, which are amphichiral.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 0, 12, 5, 83, 0, 808, 47
Offset: 1

Views

Author

David Garber, Sep 20 2005

Keywords

Comments

a(n)=0 for n=3 (mod 4)

Links

A110886 Number of signed weighted Euler trees with total weight n (associated to even switching classes of matrices of order 2n).

Original entry on oeis.org

1, 1, 3, 8, 27, 104, 436, 1930, 8871, 41916, 202300, 992942, 4940912, 24867870, 126371426, 647494746, 3341341155, 17350565376, 90593056624, 475333630402, 2504959102224, 13252904123786, 70366654738470, 374824160997086
Offset: 0

Views

Author

David Garber, Sep 19 2005

Keywords

Examples

			a(5) = 104. (1, 3, 8, 27) dot (1, 2, 5, 19) = 77; then 104 = a(4) + 77 = 27 + 77.

Links

Programs

  • Maple
    G:=(3*(1-z)-sqrt((1-z)*(1-5*z-4*z^2)))/2/(1-z): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..27); # Emeric Deutsch, Dec 31 2006
  • Mathematica
    CoefficientList[Series[(3*(1-x)-Sqrt[(1-x)*(1-5*x-4*x^2)])/2/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
a[n_] := Sum[(Binomial[2*k-2, k-1]*Sum[Binomial[k, n-k-i]*Binomial[k+i-1, k-1], {i, 0, n-k}])/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 24 2013, after Vladimir Kruchinin *)
  • Maxima
    a(n):=sum((binomial(2*k-2,k-1)*sum(binomial(k,n-k-i)*binomial(k+i-1,k-1),i,0,n-k))/k,k,1,n); /* Vladimir Kruchinin, Jan 24 2013 */
  • PARI
    N = 66;  x = 'x + O('x^N);
gf =  ( 3*(1-x)-sqrt((1-x)*(1-5*x-4*x^2))  ) / (2*(1-x));
v = Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

Formula

G.f.: ( 3*(1-z)-sqrt((1-z)*(1-5*z-4*z^2)) ) / (2*(1-z)).
a(n) = 2 + Sum_{k=1..n-1} a(n-k)*a(k). - Benoit Cloitre, Jul 27 2008
Recurrence: n*a(n) = 2*(3*n-4)*a(n-1) - (n+2)*a(n-2) - 2*(2*n-7)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ sqrt(41-3*sqrt(41))*((5+sqrt(41))/2)^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012
a(n) = Sum_{k=1..n} (binomial(2*k-2, k-1)*Sum_{i=0..n-k} binomial(k, n-k-i)*binomial(k+i-1, k-1)/k), n > 0, a(0)=1. - Vladimir Kruchinin, Jan 24 2013
a(n+1) starting (1, 3, ...) = (first n terms) dot product (first n difference terms), added to a(n). - Gary W. Adamson, May 20 2013

Extensions

More terms from Emeric Deutsch, Dec 31 2006

A076093 Squares arising in A076991.

Original entry on oeis.org

9, 16, 25, 36, 49, 64, 81, 64, 81, 100, 121, 81, 64, 49, 64, 100, 121, 144, 81, 64, 36, 49, 64, 81, 64, 81, 100, 169, 196, 225, 196, 225, 256, 324, 289, 196, 121, 81, 100, 81, 144, 121, 144, 169, 196, 169, 196, 225, 289, 324, 289, 324, 361, 484, 441, 289
Offset: 1

Views

Author

David Garber, Nov 02 2002

Keywords

Links

Crossrefs

Cf. A076991, A077396.

A077396 Square roots of squares arising in A076991.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 9, 8, 7, 8, 10, 11, 12, 9, 8, 6, 7, 8, 9, 8, 9, 10, 13, 14, 15, 14, 15, 16, 18, 17, 14, 11, 9, 10, 9, 12, 11, 12, 13, 14, 13, 14, 15, 17, 18, 17, 18, 19, 22, 21, 17, 14, 13, 12, 11, 10, 11, 13, 14, 15, 12, 13, 15, 18, 19, 20, 21, 19, 16, 13, 14, 16
Offset: 1

Views

Author

David Garber, Nov 03 2002

Keywords

Links

Crossrefs

Cf. A076093, A076991.

A073946 Squares k such that k + pi(k) is a prime.

Original entry on oeis.org

9, 36, 81, 121, 361, 625, 961, 3136, 6724, 8281, 9604, 10609, 12996, 13225, 19881, 25281, 38025, 39204, 40000, 43264, 44944, 45796, 47961, 60516, 64009, 79524, 80089, 80656, 83521, 86436, 90000, 93636, 103684, 117649, 121801, 129600
Offset: 1

Views

Author

David Garber, Nov 13 2002

Keywords

Examples

			a(1)=9, since 9 is a square, pi(9)=4 and 9+4=13 is a prime.

Links

Crossrefs

This sequence is a subsequence of sequence A077510. The corresponding sequence of primes is A113943 and the square roots of the original sequence is A113944.

Programs

  • Maple
    select(t -> isprime(t + numtheory:-pi(t)), [seq(i^2,i=1..1000)]); # Robert Israel, Mar 21 2017
  • Mathematica
    Select[Range[1000]^2, PrimeQ[# + PrimePi[#]] &] (* Indranil Ghosh, Mar 21 2017 *)
  • PARI
    v=vector(1000);
for(n=1, 1000, v[n] = n^2);
for(n=1, 1000, if(isprime(v[n] + primepi(v[n])), print1(v[n],", "))) \\ Indranil Ghosh, Mar 21 2017
  • Python
    from sympy import primepi, isprime
N = (x**2 for x in range(1, 1001))
print([n for n in N if isprime(n + primepi(n))]) # Indranil Ghosh, Mar 21 2017

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

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