cp's OEIS Frontend

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.

Showing 1-8 of 8 results.

A075381 Fixed points of A075380.

Original entry on oeis.org

1, 4, 14, 26, 34, 41, 51, 64, 71, 77, 87, 90, 93, 111, 115, 123, 132, 134, 141, 144, 146, 149, 159, 162, 165, 168, 171, 173, 177, 180, 190, 197, 219, 223, 231, 240, 242, 258, 260, 265, 266, 273, 276, 278, 287, 290, 293, 305, 317, 330, 332, 348, 350, 365, 371
Offset: 1

Views

Author

Amarnath Murthy, Sep 22 2002

Keywords

Comments

A075380(a(n))=a(n); A167901(a(n))=a(n); A167902(a(n))=a(n). [Reinhard Zumkeller, Nov 15 2009]

Links

Crossrefs

Cf. A075380.

Programs

  • PARI
    print1(1, ", "); v=[1]; n=1; while(n<10^3, if(!issquarefree(n+v[#v])&&!vecsearch(vecsort(v), n), v=concat(v, n); if(n==#v, print1(n, ", ")); n=0); n++) \\ Derek Orr, Jun 09 2015

Extensions

More terms from David Wasserman, Jan 17 2005

A167901 A075380(A075380(n)).

Original entry on oeis.org

1, 5, 8, 4, 6, 13, 3, 10, 9, 11, 2, 12, 7, 14, 17, 25, 20, 23, 18, 19, 15, 24, 21, 22, 16, 26, 32, 33, 27, 31, 28, 30, 29, 34, 40, 35, 38, 39, 37, 36, 41, 45, 49, 42, 44, 48, 43, 50, 47, 46, 51, 55, 59, 54, 63, 62, 57, 52, 60, 56, 58, 53, 61, 64, 70, 65, 68, 69, 67, 66, 71, 74
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 15 2009

Keywords

Comments

Permutation of positive integers;
a(A075381(n)) = A075381(n);
a(A167902(n)) = A167902(a(n)) = A075380(n).

Links

A167902 Inverse integer permutation to A075380.

Original entry on oeis.org

1, 7, 2, 4, 3, 8, 11, 5, 12, 6, 13, 9, 10, 14, 19, 24, 18, 15, 21, 23, 20, 16, 17, 25, 22, 26, 31, 27, 30, 33, 29, 28, 32, 34, 37, 39, 36, 35, 40, 38, 41, 47, 42, 49, 43, 48, 44, 50, 45, 46, 51, 62, 52, 57, 53, 61, 54, 56, 55, 63, 60, 58, 59, 64, 67, 69, 66, 65, 70, 68, 71, 75
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 15 2009

Keywords

Comments

a(A075381(n)) = A075381(n);
a(A075380(n)) = A075380(a(n)) = n;
a(A167901(n)) = A167901(a(n)) = A121878(n).

Links

A167903 A075380(n) + A075380(n+1).

Original entry on oeis.org

4, 8, 9, 12, 18, 12, 8, 18, 25, 20, 16, 20, 25, 32, 40, 45, 40, 32, 36, 40, 44, 45, 36, 40, 50, 54, 60, 63, 60, 56, 60, 63, 64, 72, 75, 72, 75, 76, 75, 80, 84, 88, 92, 96, 99, 92, 88, 90, 92, 99, 104, 108, 112, 116, 117, 112, 116, 125, 124, 117, 108, 112, 124, 132, 135, 132
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 15 2009

Keywords

Comments

A008966(a(n)) = 0 by definition of A075380.

Crossrefs

Cf. A167907.

A121878 a(1)=1. a(n) = the smallest positive integer not occurring earlier in the sequence such that a(n-1)+a(n) is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 12, 11, 15, 14, 16, 13, 17, 18, 19, 20, 21, 22, 24, 23, 28, 25, 26, 27, 30, 29, 32, 33, 34, 31, 35, 36, 37, 40, 38, 39, 43, 42, 41, 44, 45, 46, 47, 48, 49, 52, 50, 51, 54, 53, 56, 55, 58, 57, 61, 62, 60, 59, 63, 64, 65, 66, 67, 70, 68, 69, 72, 71
Offset: 1

Views

Author

Leroy Quet, Aug 31 2006

Keywords

Comments

Inverse: A167905; A167904(n) = a(a(n)). [Reinhard Zumkeller, Nov 15 2009]
Conjectured to be a permutation of the natural numbers. - Derek Orr, Jun 01 2015

Examples

			9,10,11,12,... are the positive integers not occurring among the first 8 terms of the sequence. a(8) + 9 = 16, which is not squarefree. a(8) + 10 = 17, which is squarefree. So a(9) = 10.

Links

Crossrefs

Cf. A167907, A075380. [Reinhard Zumkeller, Nov 15 2009]

Programs

  • Mathematica
    f[s_] := Block[{k = 1},While[MemberQ[s, k] || Max @@ Last /@ FactorInteger[(s[[ -1]] + k)] > 1, k++ ]; Append[s, k]]; Nest[f, {1}, 75] (* Ray Chandler, Sep 06 2006 *)
  • PARI
    v=[1];n=1;while(n<100,if(issquarefree(v[#v]+n)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v \\ Derek Orr, Jun 01 2015

Extensions

Extended by Ray Chandler, Sep 06 2006

A285296 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.

Original entry on oeis.org

1, 4, 2, 6, 3, 8, 5, 9, 7, 12, 10, 14, 16, 11, 18, 13, 20, 15, 21, 24, 17, 25, 19, 27, 22, 26, 28, 23, 32, 29, 36, 30, 33, 39, 40, 31, 44, 34, 38, 42, 35, 45, 37, 48, 41, 49, 43, 50, 46, 52, 47, 54, 51, 56, 53, 60, 55, 63, 57, 64, 58, 62, 66, 68, 59, 72, 61
Offset: 1

Views

Author

Rémy Sigrist, Apr 16 2017

Keywords

Comments

The sequence can always be extended with a number that is not squarefree (say a multiple of 4); after a term that is not squarefree, we can extend the sequence with the least unused number; as there are infinitely many multiples of 4, this sequence is a permutation of the natural numbers (with inverse A285297).
Conjecturally, a(n) ~ n.
This sequence has similarities with A075380: here we consider the product of consecutive terms, there the sum of consecutive terms.
For any k>0, let b_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^k for some prime p; in particular we have:
- b_1 = A000027 (the natural numbers),
- b_2 = a (this sequence),
- b_3 = A285299,
- b_4 = A285386,
- b_5 = A285417.
For any k>0, b_k is a permutation of the natural numbers.
For any k>0, b_k(1)=1 and b_k(2)=2^k.
Graphically, the sequences from b_2 to b_5 differ.

Examples

			The first terms, alongside the primes p such that p^2 divides a(n)*a(n+1), are:
n       a(n)    p
--      ----    -
1       1       2
2       4       2
3       2       2
4       6       3
5       3       2
6       8       2
7       5       3
8       9       3
9       7       2
10      12      2
11      10      2
12      14      2
13      16      2
14      11      3
15      18      3
16      13      2
17      20      2, 5
18      15      3
19      21      2, 3
20      24      2

Links

Crossrefs

Cf. A000027, A075380, A285297 (inverse).

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

Views

Author

Derek Orr, Jun 09 2015

Keywords

Comments

Believed to be a permutation of the natural numbers.

Links

Crossrefs

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

Programs

  • Haskell
    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
  • PARI
    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

Views

Author

Derek Orr, Jun 09 2015

Keywords

Comments

Numbers n such that A258767(n) = n.
Also fixed points of A258827. - Reinhard Zumkeller, Jun 11 2015

Links

Crossrefs

Cf. A258767, A075380, A075381, A256271, A258766.
Cf. A258827.

Programs

  • Haskell
    a258768 n = a258768_list !! (n-1)
a258768_list = [x | x <- [1..], a258767 x == x]
-- Reinhard Zumkeller, Jun 11 2015
  • PARI
    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++)
Showing 1-8 of 8 results.

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