A009388 -id:A009388 - 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 *)

A093906 Start with {2} and close under the operations XY and XY+1; sequence gives complete list of numbers that do not appear.

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 14, 15, 24, 27, 28, 29, 30, 31, 48, 49, 54, 57, 58, 59, 60, 61, 62, 63, 97, 98, 108, 109, 114, 117, 118, 119, 120, 123, 124, 127, 194, 197, 217, 218, 219, 228, 229, 237, 238, 239, 240, 241, 246, 247, 248, 249, 389, 394, 434, 439, 457, 458, 479, 482, 492, 493, 497, 498, 499, 788, 789, 878, 879, 917, 959, 984, 985, 986, 994, 997, 998, 1579, 1757, 1758, 1759, 1968, 1971, 1988, 1994, 1997, 3514, 3517, 3518, 3989, 7028, 7034, 7037, 7978, 14074, 28148

Offset: 1

Franklin T. Adams-Watters, May 25 2004

If we start with {2} and close under the operations XY-1 and XY+1, we get the complement of A014574 (midpoints of twin primes), except for the absence of 1.

Cf. A009388 and A009293.

Corrected by Don Reble, May 25 2004

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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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Programs

Haskell

Mathematica

A093906 Start with {2} and close under the operations XY and XY+1; sequence gives complete list of numbers that do not appear.

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Author

Keywords

Comments

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