A256271 -id:A256271 - cp's OEIS frontend

A258766 Fixed points in A256271.

1, 2, 3, 26, 32, 34, 37, 49, 55, 62, 64, 74, 75, 76, 77, 164, 171, 189, 224, 273, 279, 280, 285, 303, 333, 345, 356, 363, 368, 382, 399, 411, 416, 422, 429, 430, 435, 441, 453, 470, 472, 483, 494, 524, 539, 561, 566, 579, 580, 585, 603, 609, 621, 644, 662, 666, 674, 693, 704, 715, 737, 771, 777, 794, 803

Offset: 1

Derek Orr, Jun 09 2015

Numbers n such that A256271(n) = n.
From Robert Israel, Jul 16 2019: (Start)
A necessary condition for n to be in the sequence is that A256271(n)-n is even.  When A256271(n) is even, A256271(n+1) must be odd; when A256271(n) is odd, A256271(n+1) may be either even or odd, but it appears that it is nearly always even.
The result is that we have long intervals where A256271(n)-n is even (e.g. 3369 to 22635), in which members of this sequence are relatively common, and long intervals where A256271(n)-n is odd (e.g. 22636 to 67110) which contain no members of this sequence. (End)

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A121878, A167906, A256271.

Res:= 1: count:= 1: v:= 1:
Cands:= [$2..1000]:
for n from 2 do
  found:= false;
  for j from 1 to nops(Cands) do
    if numtheory:-issqrfree(v + Cands[j]^2) then
      found:= true;
      if n = Cands[j] then Res:= Res, n; count:= count+1 fi;
      v:= Cands[j]^2;
      Cands:= subsop(j=NULL, Cands);
      break
    fi
  od;
  if not found then break fi;
od:
Res; # Robert Israel, Jul 16 2019

print1(1,", ");v=[1]; n=1; while(#v<10^3, if(issquarefree(n^2+v[#v]^2)&&!vecsearch(vecsort(v), n), if(n==#v, print1(n, ", ")); n=0); n++)

A258767 With a(1) = 1, a(n) is the smallest number not already in the sequence such that a(n)^2 + a(n-1)^2 is not squarefree.

Original entry on oeis.org

1, 7, 14, 2, 4, 3, 6, 8, 10, 5, 12, 9, 13, 16, 18, 15, 20, 21, 22, 11, 23, 36, 24, 26, 28, 29, 47, 46, 30, 25, 35, 40, 32, 34, 17, 19, 33, 27, 31, 42, 38, 41, 37, 39, 45, 48, 44, 50, 49, 43, 51, 54, 52, 56, 58, 59, 62, 60, 55, 65, 70, 63, 57, 66, 64, 68, 72, 69, 67, 81, 75, 78, 71, 53, 79, 97, 96, 74

Offset: 1

Derek Orr, Jun 09 2015

nonn

Believed to be a permutation of the natural numbers.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256271, A075380.
Cf. A258768 (fixed points).
Cf. A008966, A258827 (putative inverse).

import Data.List (delete)
a258767 n = a258767_list !! (n-1)
a258767_list = 1 : f 1 [2..] where
   f x zs = g zs where
     g (y:ys) | a008966 (x^2 + y^2) == 1 = g ys
              | otherwise = y : f y (delete y zs)
-- Reinhard Zumkeller, Jun 11 2015

v=[1]; n=1; while(n<100, if(!issquarefree(n^2+v[#v]^2)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v

A258768 Fixed points in A258767.

Original entry on oeis.org

1, 8, 13, 34, 45, 49, 51, 80, 86, 92, 98, 146, 163, 164, 206, 216, 266, 279, 303, 312, 333, 337, 348, 356, 359, 371, 387, 388, 398, 406, 421, 432, 445, 460, 463, 465, 509, 517, 533, 536, 546, 548, 572, 576, 585, 602, 607, 612, 624, 638, 658, 666, 669, 675, 688, 704, 711, 734, 744, 765, 771, 801, 810, 814

Offset: 1

Derek Orr, Jun 09 2015

nonn

Numbers n such that A258767(n) = n.

Also fixed points of A258827. - Reinhard Zumkeller, Jun 11 2015

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

Cf. A258767, A075380, A075381, A256271, A258766.

Cf. A258827.

a258768 n = a258768_list !! (n-1)
a258768_list = [x | x <- [1..], a258767 x == x]
-- Reinhard Zumkeller, Jun 11 2015

print1(1, ", "); v=[1]; n=1; while(#v<10^3, if(!issquarefree(n^2+v[#v]^2)&&!vecsearch(vecsort(v), n), v=concat(v, n); if(n==#v, print1(n, ", ")); n=0); n++)

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A258766 Fixed points in A256271.

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A258767 With a(1) = 1, a(n) is the smallest number not already in the sequence such that a(n)^2 + a(n-1)^2 is not squarefree.

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A258768 Fixed points in A258767.

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Haskell

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