A009299 -id:A009299 - cp's OEIS frontend

A192476 Monotonic ordering of set S generated by these rules: if x and y are in S then x^2 + y^2 is in S, and 1 is in S.

1, 2, 5, 8, 26, 29, 50, 65, 68, 89, 128, 677, 680, 701, 740, 842, 845, 866, 905, 1352, 1517, 1682, 2501, 2504, 2525, 2564, 3176, 3341, 4226, 4229, 4250, 4289, 4625, 4628, 4649, 4688, 4901, 5000, 5066, 5300, 5465, 6725, 7124, 7922, 7925, 7946, 7985

Offset: 1

Clark Kimberling, Jul 01 2011

nonn

Let N denote the positive integers, and suppose that f(x,y): N x N->N. Let "start" denote a subset of N, and let S be the set of numbers defined by these rules: if x and y are in S, then f(x,y) is in S, and "start" is a subset of S. The monotonic increasing ordering of S is a sequence:
A192476:  f(x,y)=x^2+y^2, start={1}
A003586:  f(x,y)=x*y, start={1,2,3}
A051037:  f(x,y)=x*y, start={1,2,3,5}
A002473:  f(x,y)=x*y, start={1,2,3,5,7}
A003592:  f(x,y)=x*y, start={2,5}
A009293:  f(x,y)=x*y+1, start={2}
A009388:  f(x,y)=x*y-1, start={2}
A009299:  f(x,y)=x*y+2, start={3}
A192518:  f(x,y)=(x+1)(y+1), start={2}
A192519:  f(x,y)=floor(x*y/2), start={3}
A192520:  f(x,y)=floor(x*y/2), start={5}
A192521:  f(x,y)=floor((x+1)(y+1)/2), start={2}
A192522:  f(x,y)=floor((x-1)(y-1)/2), start={5}
A192523:  f(x,y)=2x*y-x-y, start={2}
A192525:  f(x,y)=2x*y-x-y, start={3}
A192524:  f(x,y)=4x*y-x-y, start={1}
A192528:  f(x,y)=5x*y-x-y, start={1}
A192529:  f(x,y)=3x*y-x-y, start={2}
A192531:  f(x,y)=3x*y-2x-2y, start={2}
A192533:  f(x,y)=x^2+y^2-x*y, start={2}
A192535:  f(x,y)=x^2+y^2+x*y, start={1}
A192536:  f(x,y)=x^2+y^2-floor(x*y/2), start={1}
A192537:  f(x,y)=x^2+y^2-x*y/2, start={2}
A192539:  f(x,y)=2x*y+floor(x*y/2), start={1}

			1^2+1^2=2, 1^2+2^2=5, 2^2+2^2=8, 1^2+5^2=26.

Reinhard Zumkeller, Table of n, a(n) for n = 1..6171

Cf. A009293, A008318.

import Data.Set (singleton, deleteFindMin, insert)
a192476 n = a192476_list !! (n-1)
a192476_list = f [1] (singleton 1) where
   f xs s =
     m : f xs' (foldl (flip insert) s' (map (+ m^2) (map (^ 2) xs')))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

start = {1}; f[x_, y_] :=  x^2 + y^2  (* start is a subset of t, and if x,y are in t then f(x,y) is in t. *)
b[z_] :=  Block[{w = z}, Select[Union[Flatten[AppendTo[w, Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # < 30000 &]];
t = FixedPoint[b, start] (* A192476 *)
Differences[t]
(* based on program by Robert G. Wilson v at A009293 *)

A009293 If a, b in sequence, so is a*b+1.

Original entry on oeis.org

2, 5, 11, 23, 26, 47, 53, 56, 95, 107, 113, 116, 122, 131, 191, 215, 227, 233, 236, 245, 254, 263, 266, 281, 287, 383, 431, 455, 467, 473, 476, 491, 509, 518, 527, 530, 533, 536, 563, 566, 575, 581, 584, 599, 611, 617, 656, 677, 767, 863, 911, 935, 947, 953, 956, 983

Offset: 1

David W. Wilson

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Reinhard Zumkeller)

Cf. A009388, A009299, A192476.

import Data.Set (singleton, deleteFindMin, insert)
a009293 n = a009293_list !! (n-1)
a009293_list = f [2] (singleton 2) where
   f xs s = m : f xs' (foldl (flip insert) s' (map ((+ 1) . (* m)) xs'))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

f[l_] := Block[{k = l}, Select[ Union[ Flatten[ AppendTo[k, Table[ k[[i]]*k[[j]] + 1, {i, 1, Length[k]}, {j, 1, i}]]]], # < 1000 &]]; NestList[f, {2}, 7][[ -1]] (* Robert G. Wilson v, May 23 2004 *)

A009388 If a, b in sequence, so is a*b-1.

Original entry on oeis.org

2, 3, 5, 8, 9, 14, 15, 17, 23, 24, 26, 27, 29, 33, 39, 41, 44, 45, 47, 50, 51, 53, 57, 63, 65, 68, 69, 71, 74, 77, 80, 81, 84, 86, 87, 89, 93, 98, 99, 101, 105, 111, 113, 114, 116, 119, 122, 125, 129, 131, 134, 135, 137, 140, 141, 144, 147, 149, 152, 153, 158, 159, 161, 164

Offset: 1

David W. Wilson

nonn

All terms are congruent to 0 or 2 mod 3. It follows that no three consecutive integers are in the sequence. - Franklin T. Adams-Watters, Aug 31 2016, conjectured by David W. Wilson.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A009293. This is superset of A005659 - 1.

Cf. A009299, A192476.

import Data.Set (singleton, deleteFindMin, insert)
a009388 n = a009388_list !! (n-1)
a009388_list = f [2] (singleton 2) where
   f xs s = m : f xs' (foldl (flip insert) s' (map (pred . (* m)) xs'))
     where xs' = m : xs
           (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 15 2011

f[l_] := Block[{k = l}, Select[ Union[ Flatten[ AppendTo[k, Table[ k[[i]]*k[[j]] - 1, {i, 1, Length[k]}, {j, 1, i}]]]], # < 170 &]]; NestList[f, {2}, 6][[ -1]] (* Robert G. Wilson v, May 23 2004 *)

cp's OEIS Frontend

A192476 Monotonic ordering of set S generated by these rules: if x and y are in S then x^2 + y^2 is in S, and 1 is in S.

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A009293 If a, b in sequence, so is a*b+1.

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Author

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Programs

Haskell

Mathematica

A009388 If a, b in sequence, so is a*b-1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica