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.
s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 7}]
s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 7}]
The set S = { 1, 2, 3, 6 } gives a sum 1/1 + 1/2 + 1/3 + 1/6 = 2; exhaustive search shows that no set with a smaller maximal element can sum to 2, therefore a(2) = 6.
a(1) = 1, S = { 1 }
a(2) = 6, S = { 1 2 3 6 }
a(3) = 24, S = { 1 2 3 4 5 6 8 9 10 15 18 20 24 }
a(4) = 65, S = { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24 26 27 28 30 33 35 36 40 42 45 48 52 54 56 60 63 65 }
a(5) = 184, using this set: { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 32 33 34 35 36 38 39 40 42 44 45 48 50 51 52 54 55 56 58 60 62 63 65 66 68 69 70 72 75 76 77 78 80 81 84 85 87 88 90 91 92 93 95 96 99 102 104 105 108 110 112 114 115 116 117 126 130 133 136 138 140 143 144 145 150 152 153 154 155 156 161 162 165 168 170 171 174 175 176 180 184 }
a(6) = 469, using the set: { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 60 61 62 63 64 65 66 67 68 69 70 72 74 75 76 77 78 80 81 82 84 85 86 87 88 90 91 92 93 94 95 96 98 99 100 102 104 105 106 108 110 111 112 114 115 116 117 119 120 121 122 123 124 126 128 129 130 132 133 134 135 136 138 140 141 143 144 145 147 148 150 152 153 154 155 156 159 160 161 162 164 165 168 170 171 174 175 176 177 180 182 183 184 185 186 187 188 189 190 192 195 196 198 200 201 203 204 205 207 208 209 210 212 215 216 217 220 221 222 224 225 228 230 231 232 234 238 240 242 245 246 247 248 250 252 253 255 258 259 260 261 264 266 268 270 272 273 275 276 280 282 285 286 287 288 290 294 295 297 299 300 301 304 305 306 310 312 315 319 320 322 323 324 325 328 329 330 336 340 341 344 345 348 350 351 352 354 357 360 363 364 368 370 372 375 376 377 378 380 384 385 387 390 396 400 402 405 406 407 408 413 414 416 418 424 425 429 430 432 434 435 440 442 444 448 451 455 456 460 462 465 468 469 }
s = HarmonicNumber@ Range[10^4]; Table[Position[s, k_ /; k < 2 HarmonicNumber@ n][[-1, 1]], {n, 48}] (* Michael De Vlieger, Jun 27 2017 *)
(* The following program searches for such n that f(n) <> a(n) *)
f[n_] := Floor[n^2*E^(EulerGamma + 1/n) - (1/2 + (1/6)*E^(EulerGamma))];
harmonic[n_] := Log[n] + EulerGamma + 1/(2 n) - Sum[BernoulliB[2 k]/(2 k*n^(2 k)), {k, 1, 10}];
Select[Range[100000], 2*harmonic[#] < harmonic[f[#]] &]
(* Vaclav Kotesovec, Jul 17 2017 *)
a(n) = {my(m=1); hn = sum(k=1, n, 1/k); hm = 1; until(hm > 2*hn, m++; hm+=1/m); m--;} \\ Michel Marcus, Jul 19 2017
from sympy import harmonic def a(n): hn2 = 2 * harmonic(n) m = n while harmonic(m) <= hn2: m += 1 return m - 1 print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Mar 10 2021
s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(1/3), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]
Obviously a(1)=1. If the center of gravity of one brick is placed at the end of a second brick, the length of the stack of 2 bricks is 1.5. If the c.g. of that stack is placed at the end of a third brick, the length of the stack is 1.75. Continuing, we get a stack of length 1.916666... for 4 bricks and a stack of length 2.0416666... for 5 bricks. Thus a(2)=5.
A002387[n_] := Floor[ Exp[n - EulerGamma] + 1/2]; a[n_] := A002387[2n - 2] + 1; a[1] = 1; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Dec 13 2011, after Charles R Greathouse IV *) f[n_] := k /. FindRoot[HarmonicNumber[k -1] == 2n, {k, Exp[ 2n]}, WorkingPrecision -> 100] // Ceiling; Array[f, 21, 0] (* Robert G. Wilson v, Jan 26 2017 after Jean-François Alcover in A014537 *) (* note that the index is off by one *)
[Floor(Exp(HarmonicNumber(n))*Log(HarmonicNumber(n))): n in [1..80]]; // G. C. Greubel, Jan 15 2019
a[n_] := Exp[HarmonicNumber[n]] Log[HarmonicNumber[n]] // Floor; Array[a, 64] (* Jean-François Alcover, Oct 08 2018 *)
{h(n) = sum(k=1, n, 1/k)};
vector(80, n, floor( exp(h(n))*log(h(n))) ) \\ G. C. Greubel, Jan 15 2019
[floor(exp(harmonic_number(n))*log(harmonic_number(n))) for n in (1..80)] # G. C. Greubel, Jan 15 2019
With InvH(x) being the inverse of H(x), x > 0, an asymptotic series for InvH(x) + 1/2 is u - 1/(24u) + 3/(640u^3) - 1525/(580608u^5) +-... where u = e^(x - g) and g is Euler's gamma constant.
n = 12; coeffs = InverseSeries[Exp[Series[HarmonicNumber[x - 1/2], {x, Infinity, 2n - 1}] - EulerGamma]][[3]]; Table[Numerator[coeffs[[2i - 1]]], {i, 1, n}]
With InvH(x) being the inverse of H(x), x > 0, an asymptotic series for InvH(x) + 1/2 is u - 1/(24u) + 3/(640u^3) - 1525/(580608u^5) +-... where u = e^(x - g) and g is Euler's gamma constant.
n = 12; coeffs = InverseSeries[Exp[Series[HarmonicNumber[x - 1/2], {x, Infinity, 2n - 1}] - EulerGamma]][[3]]; Table[Denominator[coeffs[[2i - 1]]], {i, 1, n}]
1/2+1/3+1/6=1
{r=1;u=[];l=1;for(n=1,99,while(setsearch(u,l),l++);m=ceil(1/r);while(setsearch(u,m),m++);print1(m",");r-=1/m;r||r=1;u=setunion(u,Set(m)))} \\ M. F. Hasler
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