Displaying 1-10 of 30 results found.
Numbers not divisible by 5.
+10
53
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87
COMMENTS
Original name was: Numbers that are congruent to {1, 2, 3, 4} mod 5.
More generally the sequence of numbers not divisible by some fixed integer m>=2 is given by a(n,m) = n-1+floor((n+m-2)/(m-1)). - Benoit Cloitre, Jul 11 2009
FORMULA
G.f.: (x+2*x^2+3*x^3+4*x^4+4*x^5+3*x^6+2*x^7+x^8)/(1-x^4)^2 (not reduced). - Len Smileya(n) = 5+a(n-4).
G.f.: x*(1+x+x^2+x^3+x^4)/((1-x)*(1-x^4)).
a(n) = a(n-1) + a(n-4) - a(n-5), n>5.
a(n) = (10*n-5-(-1)^n+2*(-1)^((2*n+5-(-1)^n)/4))/8. (End)
E.g.f.: 1 + (1/4)*(-cos(x) + (-3 + 5*x)*cosh(x) + sin(x) + (-2 + 5*x)*sinh(x)). - Stefano Spezia, Dec 01 2019
MAPLE
seq(floor((15*n-1)/12), n=1..56); # Gary Detlefs, Mar 07 2010
PROG
(PARI) a(n)= n+(n-1)\4 \\ corrected by Michel Marcus, Sep 02 2022 (Sage) [i for i in range(72) if gcd(5, i) == 1] # Zerinvary Lajos, Apr 21 2009 (Haskell)
a047201 n = a047201_list !! (n-1)
a047201_list = [x | x <- [1..], mod x 5 > 0]
Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).
+10
30
9, 5, 1, 0, 5, 6, 5, 1, 6, 2, 9, 5, 1, 5, 3, 5, 7, 2, 1, 1, 6, 4, 3, 9, 3, 3, 3, 3, 7, 9, 3, 8, 2, 1, 4, 3, 4, 0, 5, 6, 9, 8, 6, 3, 4, 1, 2, 5, 7, 5, 0, 2, 2, 2, 4, 4, 7, 3, 0, 5, 6, 4, 4, 4, 3, 0, 1, 5, 3, 1, 7, 0, 0, 8, 5, 1, 9, 3, 5, 0, 1, 7, 1, 8, 7, 9, 2, 8, 1, 0, 9, 7, 0, 8, 1, 1, 3, 8, 1
COMMENTS
Circumscribed sphere radius for a regular icosahedron with unit edges. - Stanislav Sykora, Feb 10 2014 Side length of the particular golden rhombus with diagonals 1 and phi ( A001622); area is phi/2 ( A019863). Thus, also the ratio side/(shorter diagonal) for any golden rhombus. Interior angles of a golden rhombus are always A105199 and A137218. - Rick L. Shepherd, Apr 10 2017 An algebraic number of degree 4; minimal polynomial is 16x^4 - 20x^2 + 5, which has these smaller, other solutions (conjugates): - A019881 < - A019845 < A019845 (sine of 36 degrees). - Rick L. Shepherd, Apr 11 2017 This is length ratio of one half of any diagonal in the regular pentagon and the circumscribing radius. - Wolfdieter Lang, Jan 07 2018 This gives the imaginary part of one of the members of a conjugate pair of roots of x^5 - 1, with real part (-1 + phi)/2 = A019827, where phi = A001622. A member of the other conjugte pair of roots is (-phi + sqrt(3 - phi)*i)/2 = (- A001622 + A182007*i)/2 = - A001622/2 + A019845*i. - Wolfdieter Lang, Aug 30 2022
REFERENCES
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.
FORMULA
Equals sqrt((5+sqrt(5))/8) = cos(Pi/10). - Zak Seidov, Nov 18 2006 Equals 2F1(13/20,7/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
EXAMPLE
0.95105651629515357211643933337938214340569863412575022244730564443015317008...
PROG
(PARI)
default(realprecision, 120);
(Magma) SetDefaultRealField(RealField(100)); Sqrt((5 + Sqrt(5))/8); // G. C. Greubel, Nov 02 2018
CROSSREFS
Cf. A001622, A019827, A102208, A179290 (inverse), A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A182007.
Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.
+10
26
7, 5, 5, 7, 6, 1, 3, 1, 4, 0, 7, 6, 1, 7, 0, 7, 3, 0, 4, 8, 0, 1, 3, 3, 7, 0, 2, 0, 2, 5, 0, 0, 1, 3, 9, 2, 6, 3, 8, 4, 4, 4, 7, 8, 8, 8, 9, 3, 5, 6, 1, 0, 5, 9, 2, 2, 9, 5, 8, 2, 8, 9, 2, 0, 3, 9, 1, 0, 6, 8, 4, 5, 2, 2, 1, 9, 4, 8, 2, 6, 2, 0, 6, 3, 5, 6, 0, 4, 9, 4, 7, 6, 0, 8, 6, 8, 2, 7, 0, 4, 1, 1, 9, 3, 1
COMMENTS
Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.
REFERENCES
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.
FORMULA
Equals sqrt(42 + 18*sqrt(5))/12.
EXAMPLE
0.75576131407617073048013370202500139263844478889356105922958289203910...
MATHEMATICA
RealDigits[(Sqrt[42+18Sqrt[5]]/12), 10, 175][[1]]
Decimal expansion of radius of inscribed sphere of an icosahedron with radius of circumscribed sphere = 1.
+10
23
7, 9, 4, 6, 5, 4, 4, 7, 2, 2, 9, 1, 7, 6, 6, 1, 2, 2, 9, 5, 5, 5, 3, 0, 9, 2, 8, 3, 2, 7, 5, 9, 4, 0, 4, 2, 0, 2, 6, 5, 9, 0, 5, 8, 8, 3, 0, 9, 2, 6, 4, 8, 0, 1, 7, 5, 4, 9, 5, 5, 7, 7, 5, 0, 0, 8, 4, 3, 8, 6, 4, 4, 9, 7, 1, 7, 3, 7, 1, 1, 6, 7, 9, 3, 0, 2, 7, 2, 9, 9, 4, 8, 4, 8, 7, 0, 8, 7, 1, 3, 7, 8, 5, 2, 8
COMMENTS
Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.
FORMULA
Sqrt(75 + 30*sqrt(5))/15.
EXAMPLE
0.794654472291766122955530928327594042026590588309264801754955775008438...
MATHEMATICA
R=1; RealDigits[N[(Sqrt[75+30*Sqrt[5]]/15)*R, 175]]
EXTENSIONS
Offset corrected, keyword:cons added by R. J. Mathar, Jul 11 2010
Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.
+10
22
1, 5, 3, 8, 8, 4, 1, 7, 6, 8, 5, 8, 7, 6, 2, 6, 7, 0, 1, 2, 8, 5, 1, 4, 5, 2, 8, 8, 0, 1, 8, 4, 5, 4, 9, 1, 2, 0, 0, 3, 3, 5, 1, 0, 7, 1, 7, 6, 8, 8, 9, 6, 2, 1, 3, 5, 1, 9, 5, 7, 8, 1, 2, 5, 1, 8, 7, 4, 3, 1, 6, 4, 4, 2, 4, 7, 5, 4, 5, 4, 5, 9, 2, 2, 7, 2, 9, 6, 8, 6, 0, 8, 3, 3, 5, 5, 2, 7, 1, 7, 6, 3, 5, 9, 5
COMMENTS
Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.
Height of a regular pentagon with side length 1. - Jared Kish, Oct 16 2014 Volume of a regular decagonal prism with unit side length and height 2. - Wesley Ivan Hurt, May 04 2021
LINKS
Eric Weisstein's World of Mathematics, Pentagon
FORMULA
Equals sqrt(5+2*sqrt(5))/2.
EXAMPLE
1.53884176858762670128514528801845491200335107176889621351957812518743...
MATHEMATICA
RealDigits[Sqrt[5+2Sqrt[5]]/2, 10, 120][[1]] (* Harvey P. Dale, Jun 23 2017 *)
PROG
(PARI) sqrt(5+2*sqrt(5))/2
Decimal expansion of the volume of square cupola with edge length 1.
+10
20
1, 9, 4, 2, 8, 0, 9, 0, 4, 1, 5, 8, 2, 0, 6, 3, 3, 6, 5, 8, 6, 7, 7, 9, 2, 4, 8, 2, 8, 0, 6, 4, 6, 5, 3, 8, 5, 7, 1, 3, 1, 1, 4, 5, 8, 3, 5, 8, 4, 6, 3, 2, 0, 4, 8, 7, 8, 4, 4, 5, 3, 1, 5, 8, 6, 6, 0, 4, 8, 8, 3, 1, 8, 9, 7, 4, 7, 3, 8, 0, 2, 5, 9, 0, 0, 2, 5, 8, 3, 5, 6, 2, 1, 8, 4, 2, 7, 7, 1, 5, 1, 5, 6, 6, 7
COMMENTS
Square cupola: 12 vertices, 20 edges, and 10 faces.
Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013 Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014 2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022
FORMULA
Equals (3 + 2*sqrt(2))/3.
EXAMPLE
1.942809041582063365867792482806465385713114583584632048784453158660...
MATHEMATICA
RealDigits[N[1+(2*Sqrt[2])/3, 200]]
(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)
CROSSREFS
Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268. Cf. A131594 (decimal expansion of sqrt(2)/3).
Decimal expansion of the volume of great icosahedron with edge length 1.
+10
19
3, 1, 8, 3, 0, 5, 0, 0, 9, 3, 7, 5, 0, 8, 7, 6, 2, 6, 4, 9, 6, 1, 7, 7, 6, 3, 8, 0, 2, 8, 6, 3, 4, 9, 0, 1, 8, 9, 9, 7, 4, 2, 3, 5, 0, 1, 6, 1, 8, 6, 4, 2, 8, 1, 5, 5, 3, 7, 9, 2, 8, 1, 4, 4, 1, 2, 2, 8, 2, 9, 4, 7, 6, 5, 0, 9, 1, 4, 6, 2, 5, 2, 4, 3, 9, 9, 3, 9, 9, 6, 5, 0, 8, 8, 4, 0, 7, 1, 8, 7, 6, 2, 7, 0, 4
COMMENTS
Great icosahedron: 20 faces, 12 vertices, and 30 edges.
FORMULA
Digits of 5/12 * (3-sqrt(5)).
EXAMPLE
0.31830500937508762649617763802863490189974235016186428155379281441228294...
MATHEMATICA
RealDigits[N[5*(Sqrt[5]-3)/12, 105]][[1]]
Decimal expansion of the volume of an icosidodecahedron with edge length 1.
+10
17
1, 3, 8, 3, 5, 5, 2, 5, 9, 3, 6, 2, 4, 9, 4, 0, 4, 1, 3, 9, 8, 2, 5, 9, 9, 2, 0, 6, 1, 4, 0, 5, 2, 8, 2, 6, 6, 7, 0, 8, 1, 7, 5, 2, 0, 1, 8, 8, 9, 9, 3, 2, 2, 8, 8, 5, 4, 3, 4, 2, 0, 8, 8, 6, 1, 9, 9, 6, 4, 7, 5, 9, 5, 5, 9, 7, 3, 7, 8, 0, 5, 4, 8, 3, 4, 0, 8, 4, 0, 8, 2, 3, 7, 3, 9, 8, 8, 3, 1, 1, 2, 4, 1, 3, 6
COMMENTS
Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.
EXAMPLE
13.83552593624940413982599206140528266708175201889932288543420886199647...
MATHEMATICA
RealDigits[N[(45+17*Sqrt[5])/6, 200]]
Decimal expansion of the surface area of an icosidodecahedron with side length 1.
+10
17
2, 9, 3, 0, 5, 9, 8, 2, 8, 4, 4, 9, 1, 1, 9, 8, 9, 5, 4, 0, 7, 4, 5, 3, 7, 5, 4, 3, 6, 1, 9, 2, 6, 7, 7, 0, 2, 7, 6, 0, 2, 5, 1, 6, 3, 0, 9, 1, 7, 4, 2, 8, 3, 0, 9, 0, 7, 6, 4, 1, 7, 1, 3, 8, 1, 5, 4, 6, 0, 9, 2, 9, 9, 1, 0, 5, 1, 5, 9, 4, 9, 6, 1, 3, 9, 5, 0, 2, 5, 8, 3, 0, 4, 3, 7, 2, 9, 5, 7, 6, 4, 3, 0, 4, 6
COMMENTS
Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.
FORMULA
Sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5))))
EXAMPLE
29.3059828449119895407453754361926770276025163091742830907641713815460...
MATHEMATICA
RealDigits[N[Sqrt[30*(10+3*Sqrt[5]+Sqrt[75+30*Sqrt[5]])], 200]]
PROG
(PARI) polrootsreal(x^8 - 1200*x^6 + 324000*x^4 - 27000000*x^2 + 324000000)[8] \\ Charles R Greathouse IV, Oct 30 2023
Decimal expansion of the volume of pentagonal pyramid with edge length 1.
+10
15
3, 0, 1, 5, 0, 2, 8, 3, 2, 3, 9, 5, 8, 2, 4, 5, 7, 0, 6, 8, 3, 7, 1, 5, 5, 6, 9, 5, 3, 0, 4, 6, 9, 8, 4, 3, 1, 4, 3, 3, 5, 9, 0, 9, 8, 3, 1, 7, 1, 4, 6, 9, 0, 5, 1, 7, 7, 9, 5, 4, 0, 5, 1, 8, 9, 2, 1, 0, 5, 0, 3, 8, 5, 6, 8, 2, 4, 1, 8, 7, 0, 8, 0, 8, 9, 3, 3, 9, 3, 3, 6, 8, 2, 4, 4, 9, 2, 6, 1, 4, 5, 7, 0, 6, 2
COMMENTS
Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.
FORMULA
Digits of (5+sqrt(5))/24.
EXAMPLE
0.3015028323958245706837155695304698431433590983171469051779540518921...
MATHEMATICA
RealDigits[N[(5+Sqrt[5])/24, 200]]
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