A184676
a(n) = n + floor((n/2-1/(4*n))^2); complement of A183867.
Original entry on oeis.org
1, 2, 5, 7, 11, 14, 19, 23, 29, 34, 41, 47, 55, 62, 71, 79, 89, 98, 109, 119, 131, 142, 155, 167, 181, 194, 209, 223, 239, 254, 271, 287, 305, 322, 341, 359, 379, 398, 419, 439, 461, 482, 505, 527, 551, 574, 599, 623, 649, 674, 701, 727, 755, 782, 811, 839
Offset: 1
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[n+Floor((n/2-1/(4*n))^2): n in [1..60]]; // Vincenzo Librandi, Dec 09 2015
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seq(n+floor((n/2-1/(4*n))^2), n=1..56); # Nathaniel Johnston, Jun 26 2011
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a[n_]:=n+Floor[(n/2-1/(4n))^2];
b[n_]:=n+Floor[n^(1/2)+(n+1/2)^(1/2)];
Table[a[n],{n,1,120}] (* A184676 *)
Table[b[n],{n,1,120}] (* A183867 *)
FindLinearRecurrence[Table[a[n],{n,1,120}]]
LinearRecurrence[{2,0,-2,1},{1,2,5,7},56] (* Ray Chandler, Aug 02 2015 *)
Table[n + Floor[(n/2 - 1/(4 n))^2], {n, 60}] (* Vincenzo Librandi, Dec 09 2015 *)
Table[Ceiling[n/2] (2 + Ceiling[n/2] - Mod[n, 2]) - 1, {n, 1, 56}] (* Fred Daniel Kline, Jun 24 2016 *)
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a(n) = n+floor((n/2-1/(4*n))^2); \\ Michel Marcus, Dec 09 2015
A184675
a(n) = n + floor(sqrt(n) + sqrt(n+1)); complement of A184674.
Original entry on oeis.org
3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83
Offset: 1
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A184675:=n->n + floor(sqrt(4*n + 1)); seq(A184675(n), n=1..50); # Wesley Ivan Hurt, Mar 01 2014
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a[n_]:=n+Floor[(n/2-1/(2n))^2];
b[n_]:=n+Floor[n^(1/2)+(n+1)^(1/2)];
Table[a[n],{n,1,120}] (* A184674 *)
Table[b[n],{n,1,120}] (* A184675 *)
A265436
a(n) is the least m (1 <= m <= n) such that the set of pairs (x, y) of distinct terms from [m, n] can be ordered in such a way that the corresponding sums (x+y) and products (x*y) are monotonic.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 54, 55
Offset: 1
For n=1, the only possible interval is [1,1], the set of distinct pairs is empty, so it satisfies the desired property, hence m=1 and a(1)=1.
For n=2, the candidate interval is [1,2], the set of distinct pairs is reduced to (1,2), which satisfies the order property hence m=1 and a(2)=1.
For n=3, the candidate interval is [1,2,3], with distinct pairs (1,2), (1,3), (2,3); and with corresponding sums (3,4,5) and products (2,3,6), that are monotonically ordered, hence m=1, so a(3)=1.
For n=5, the interval [1,5] fails to produce an ordering where both sums and products follow a monotonic order. But with m=2, here is a correct ordering: (5,6), (6,8), (7,10), (7,12), (8,15), (9,20); hence m=2 and a(5)=2.
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pairs[m_, n_] := Flatten[Table[{x, y}, {x, m, n-1}, {y, x+1, n}], 1]; csum[ {x1_, y1_}, {x2_, y2_}] := x1+y1 <= x2+y2; cprod[{x1_, y1_}, {x2_, y2_}] := Which[x1 y1 < x2 y2, True, x1 y1 == x2 y2, x1+y1 <= x2+y2, True, False ]; a[1]=1; a[n_] := For[m=1, mJean-François Alcover, Dec 20 2015 *)
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vpairs(n, m, nbp) = {v = vector(nbp); k = 1; for (i=m, n-1, for (j=i+1, n, v[k] = [i, j]; k++;)); v;}
vsums(v) = vector(#v, k, v[k][1] + v[k][2]);
vprods(v) = vector(#v, k, v[k][1] * v[k][2]);
cmpp(va, vb) = {sa = va[1]+va[2]; sb = vb[1]+vb[2]; if (sa > sb, return (1)); if (sa < sb, return (-1)); pa = va[1]*va[2]; pb = vb[1]*vb[2]; pa - pb;}
isok(n, m) = {nb = n-m+1; nbp = nb*(nb-1)/2; v = vpairs (n, m, nbp); perm = vecsort(v,cmpp,1); vs = vsums(v); vp = vprods(v); vss = vector(#vs, k, vs[perm[k]]); vps = vector(#vp, k, vp[perm[k]]); (vecsort(vps) == vps) && (vecsort(vss) == vss);}
one(n, m) = {ok = 0; while (!ok, if (! isok(n, m), m++, ok=1)); m;}
lista(nn) = {m = 1; for (n=1, nn, newm = one(n, m); print1(newm, ", "); m = newm;);}
\\ Michel Marcus, Dec 09 2015
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def f1(X):
x = X
for y in range (1,X + 1): # ie 1 thru X
x = ((((((2 + y) * y) // (2 + x)) - 2) + x) // (2 + x)) + x # floor division
return x
def f0(X):
return (f1(X) + 1) - X
for x in range(1000):
print (f0(x))
# Bill McEachen, Jun 12 2024 (via the QSYNT link)
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