Calendar for Sequence of the Day in June

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Template:Sequence of the Day for June 1 A118905: Sum of legs of Pythagorean triangles (without multiple entries). { 7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, ... } Are these just the positive multiples of A001132? Richard Choulet comments: A001132 is exactly formed by the prime numbers of A118905 : in fact at first every prime p of A118905 is p=u^2-v^2+2uv, with for example u odd and v even so that : p-1=4u'(u'+1)-4v'(2u'+1-v') when u=2u'+1 and v=2v'. u'(u'+1) is even and v'(2u'+1-v') is always even. At second hand if p=8k+-1, p has the shape x^2-2y^2 ; letting u=x-y and v=y, comes p=(x-y)^2-y^2+2(x-y)y=u^2-v^2+2uv so p is a sum of the two legs of a pythagorean triangle.	Template:Sequence of the Day for June 2 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 3 A060006: Decimal expansion of the real root of $\scriptstyle x^{3}-x-1\,$. 1.324717957244746... This number has on occasion been called the "silver constant," by way of analogy to the golden ratio $\scriptstyle \phi \,$, for which we can see that $\scriptstyle \phi ^{2}-\phi -1\,=\,0\,$.	Template:Sequence of the Day for June 4 A051762: Decimal expansion of the polygon-circumscribing constant $\scriptstyle \prod _{n=3}^{\infty }{\frac {1}{\cos({\frac {\pi }{n}})}}\,$. 8.700036625208... Here is the geometric interpretation: Begin with a unit circle. Circumscribe an equilateral triangle and then circumscribe a circle. Circumscribe a square and then circumscribe a circle. Circumscribe a regular pentagon and then circumscribe a circle, etc. The circles have radii which converge to this value. The reciprocal of the polygon-circumscribing constant gives the polygon-inscribing constant (A085365) (Cf. Inscribed polygons) 1 / 8.700036625208... = 0.1149420448532...	Template:Sequence of the Day for June 5 A654321: Sequence title. { ... 6, 5, 4, 3, 2, 7, 1, ... } A sentence or two of additional details.	Template:Sequence of the Day for June 6 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 7 A021256: Decimal expansion of $\scriptstyle {\frac {1}{252}}\,$. 0.00396825396825... Note that $\scriptstyle \zeta (-5)\,=\,{\frac {-1}{252}}\,$ (referring to the Riemann zeta function).
Template:Sequence of the Day for June 8 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 9 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 10 A045917: Number of decompositions of $\scriptstyle 2n\,$ into unordered sums of two primes. { 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, ... } The term "Goldbach's comet" generally refers to a graph of this sequence.	Template:Sequence of the Day for June 11 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 12 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 13 A023036: Smallest positive even integer that is an unordered sum of two primes in exactly $\scriptstyle n\,$ ways. { 2, 4, 10, 22, 34, 48, 60, 78, 84, 90, ... } Note that the offset of this sequence is 0. There are 0 ways to express 2 as an unordered sum of two primes. If the Goldbach conjecture is true, there is no other even integer with this property.	Template:Sequence of the Day for June 14 A007507: Decimal expansion of Gelfond-Schneider constant $\scriptstyle 2^{\sqrt {2}}\,$. 2.6651441426902... This is a transcendental number, a fact that David Hilbert thought would be more difficult to prove than either the Riemann hypothesis or Fermat's last theorem. Rodion Kuzmin proved that this number is transcendental in 1930, whereas Fermat's last theorem was not proven until 1994 and the Riemann hypothesis, widely believed to be true, remains unproven.
Template:Sequence of the Day for June 15 A004002: Benford numbers: $\scriptstyle e\uparrow \uparrow n\,$ rounded to nearest integer. { 1, 3, 15, 3814279, ... } These correspond to 1, 2.718281..., 15.15426... and 3814279.10476... The next term is approximately 2.331504399$\times 10^{1656520}$. See also: Knuth's arrow notation Tetration	Template:Sequence of the Day for June 16 A000954: Conjecturally largest even integer which is an unordered sum of two primes in exactly $\scriptstyle n\,$ ways. { 2, 12, 68, 128, 152, 188, 332, 398, 368, ... } The Goldbach conjecture has been checked for millions of numbers, and yet, we can't say that this sequence is completely correct. The most dramatic possible upheaval here would be to have to replace the initial 2 with some very large number.	Template:Sequence of the Day for June 17 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 18 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 19 A137245: Decimal expansion of $\scriptstyle \sum _{i=1}^{\infty }{\frac {1}{p_{i}\log p_{i}}}\,$ over the primes. 1.63661632335... Daniel Forgues wonders if it's a coincidence or not that this number is close to $\scriptstyle 1+{\frac {2}{\pi }}\,=\,1.636619772367581...\,$ (cf. A060294 for Buffon's constant $\scriptstyle {\frac {2}{\pi }}\,$).	Template:Sequence of the Day for June 20 A010527: Decimal expansion of $\scriptstyle {\frac {\sqrt {3}}{2}}\,$. { 0.8660254037844... } This is the sine of 60 degrees, or the cosine of 30 degrees. But it is also the imaginary part of a complex cubic root of –1, for, as you can verify, $\left({\frac {1}{2}}+{\frac {\sqrt {-3}}{2}}\right)^{3}=-1.\,$	Template:Sequence of the Day for June 21 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...
Template:Sequence of the Day for June 22 A060294: Decimal expansion of Buffon's constant $\scriptstyle {\frac {2}{\pi }}\,$. 0.63661977236758... Buffon's needle problem: Theorem. The probability $\scriptstyle P(l,\,d)\,$ that a needle of length $\scriptstyle l\,$ will randomly land on a line, given a floor with equally spaced parallel lines at a distance $\scriptstyle d\,\geq \,l\,$ apart, is $\scriptstyle P(l,\,d)\,=\,{\frac {2}{\pi }}\cdot {\frac {l}{d}}\,$. Proof: (assuming that the angle and the position of the fallen needle are independently and uniformly random) If the needle always fell perpendicular (angle $\scriptstyle \theta \,=\,{\frac {\pi }{2}}\,$ radians) to the parallel lines, we would have $\scriptstyle P_{\perp }(l,\,d)\,=\,{\frac {l}{d}}\,$. So we have $P(l,d)\,=\,\left(\int _{0}^{\pi }\sin \theta \cdot {\frac {d\theta }{\pi }}\right)\cdot P_{\perp }(l,d)$ $=\left({\frac {-[\cos \theta ]_{0}^{\pi }}{\pi }}\right)\cdot {\frac {l}{d}}={\frac {2}{\pi }}\cdot {\frac {l}{d}}\,$	Template:Sequence of the Day for June 23 A028444: Busy Beaver sequence: maximal number of 1's that an $\scriptstyle n\,$-state, $\scriptstyle n\,\geq \,0\,$, Turing machine can print on an initially blank tape before halting. { 0, 1, 4, 6, 13, ... } The next term is at least 4098, and the one after that at least $1.29\times 10^{865}$. June 23, 2012 was the 100th birthday of Alan Mathison Turing (23 June 1912 – 7 June 1954).	Template:Sequence of the Day for June 24 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 25 A008908: Number of halving and tripling steps to reach 1 in the 3x + 1 problem. { 1, 2, 8, 3, 6, 9, 17, 4, 20, 7, 15, 10, 10, ... } What is so fascinating about this sequence is that it looks so random, and yet there is a precise method to determine its terms. Though the question remains whether or not there is a term that is actually null (that is to say, that 1 is never reached), due to either the process entering a loop (of finite length) after a finite number of steps; the process growing without bounds.	Template:Sequence of the Day for June 26 A655555: Sequence name { 2, 1, 3, 4, 7, 6, 5, ... } More details, two sentences maybe, two paragraph tops...	Template:Sequence of the Day for June 27 A002378: Pronic numbers $n^{2}+n$ ... 1722, 1806, 1892, 1980, 2070, 2162, 2256, ... The mathematician Augustus De Morgan was born on this date in 1806. When he turned 43 in 1849, he noticed that his age happened to be the square root of the year. The reason this was the case is that the year of his birth is a pronic number. These are numbers of the form $n^{2}+n$, but they can also be expressed as $n^{2}-n$ (with a different offset, of course). Thus 1849 = 432, and 1806 = 432 – 43. People born in 1892 may have noticed a similar situation applied when they turned 44, but among those who are still with us today, people born in 1980 who are still alive in 2025 will then be able to say that their age is the square root of the year.	Template:Sequence of the Day for June 28 A058291: Continued fraction expansion of 2 pi, ($\scriptstyle 2\pi \,=\,6.283185\dots \,$) ${6+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{7+{\cfrac {1}{2+{\cfrac {1}{146+\ddots }}}}}}}}}}}}}\,$ On the argument that $\scriptstyle \tau \,=\,2\pi \,$ is a more important constant than $\scriptstyle \pi \,$, there is a campaign going on to have today acknowledged as "Tau Day" (like March 14 is sometimes called "Pi Day"). However, neither celebration is official.
Template:Sequence of the Day for June 29 A090684: Primes of the form $\scriptstyle 8n^{2}-1\,$. { 7, 31, 71, 127, 199, 647, 967, 1151, 1567, 2311, 2591, 2887, 3527, 4231, ... } 221 223 225 227 229 231 233 235 237 239 241 219 145 147 149 151 153 155 157 159 161 163 217 143 85 87 89 91 93 95 97 99 165 215 141 83 41 43 45 47 49 51 101 167 213 139 81 39 13 15 17 19 53 103 169 211 137 79 37 11 1 3 21 55 105 171 209 135 77 35 9 7 5 23 57 107 173 207 133 75 33 31 29 27 25 59 109 175 205 131 73 71 69 67 65 63 61 111 177 203 129 127 125 123 121 119 117 115 113 179 201 199 197 195 193 191 189 187 185 183 181 In the odd number variant of the Ulam spiral, unimpeded by the even numbers, the prime numbers can line up in horizontal and vertical lines. But there are still noticeable diagonal lines of primes, and these primes fall on one such diagonal.	Template:Sequence of the Day for June 30 A065434: Imaginary part of the second nontrivial zero of the Riemann zeta function 21.02203963877... The first nontrivial zero gets all the attention. This second number, having a real part of $\scriptstyle {\frac {1}{2}}\,$, is also interesting and worth a little fame.