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 1-4 of 4 results.

A257120 Position of first occurrence of n in A256918.

Original entry on oeis.org

1, 4, 3, 5, 7, 11, 29, 13, 10, 15, 122, 19, 200, 31, 24, 18, 299, 22, 824, 16, 33, 124, 945, 41, 26, 202, 21, 37, 3093, 45, 7059, 40, 126, 301, 53, 66, 13527, 826, 204, 42, 28658, 36, 40810, 130, 57, 947, 46657, 64, 80, 44, 303, 208, 57449, 49, 133, 38, 828
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 25 2015

Keywords

Comments

A256918(a(n)) = n and A256918(m) != n for m < a(n);
GCD(A257218(a(n)),A257218(a(n)+1)) = n;
if A257218 is a permutation, then also this sequence is a permutation;
a(n) < A257475(n); a(n) = A257475(n) - A257478(n).
a(n) >= A257122(n)-1. - Ivan Neretin, May 02 2015

Examples

			See A257478.

Links

Crossrefs

Cf. A256918, A257475, A257478, A257218.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a257120 = (+ 1) . fromJust . (`elemIndex` a256918_list)

Extensions

a(41)-a(57) from Hiroaki Yamanouchi, May 03 2015

A257475 Position of second and last occurrence of n in A256918.

Original entry on oeis.org

2, 6, 9, 12, 8, 14, 30, 17, 20, 28, 123, 23, 201, 32, 25, 39, 300, 46, 825, 43, 34, 125, 946, 51, 27, 203, 47, 52, 3094, 56, 7060, 62, 127, 302, 54, 71, 13528, 827, 205, 61, 28659, 79, 40811, 132, 58, 948, 46658, 65, 81, 99, 304, 210, 57450, 69, 134, 77, 829
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 25 2015

Keywords

Comments

GCD(A257218(a(n)),A257218(a(n)-1)) = n;
a(n) > A257120(n); a(n) = A257120 + A257478(n).

Examples

			Cf. A257478.

Links

Crossrefs

Cf. A256918, A257120, A257478, A257218.

Programs

  • Haskell
    a257475 n = f 1 a256918_list where
   f i (u:us) = (if u == n then g else f) (i + 1) us
   g j (v:vs) =  if v == n then j else g  (j + 1) vs

Extensions

a(38)-a(57) from Hiroaki Yamanouchi, May 03 2015

A257478 Distance of positions of first and last occurrence of n in A256918.

Original entry on oeis.org

1, 2, 6, 7, 1, 3, 1, 4, 10, 13, 1, 4, 1, 1, 1, 21, 1, 24, 1, 27, 1, 1, 1, 10, 1, 1, 26, 15, 1, 11, 1, 22, 1, 1, 1, 5, 1, 1, 1, 19, 1, 43, 1, 2, 1, 1, 1, 1, 1, 55, 1, 2, 1, 20, 1, 39, 1, 1, 1, 13, 1, 1, 32, 42, 1, 11, 1, 2, 1, 67, 1, 26
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 25 2015

Keywords

Comments

a(n) = A257475(n) - A257120(n).

Examples

			Let w(n) = A257218(n),
u(n) = A257120(n), xx'(n) = (w(u(n)),w(u(n)+1)),
v(n) = A257475(n), yy'(n) = (w(v(n)),w(v(n)+1)):
. ----+------+------+------++--------------+------------+---------+
.   n | a(n) | u(n) | v(n) ||       xx'(n) |     yy'(n) | ... gcd |
. ----+------+------+------++--------------+------------+---------+
.   1 |    1 |    1 |    2 ||     (1,   2) |   (2,   3) |       1 |
.   2 |    2 |    4 |    6 ||     (6,   4) |   (8,  10) |       2 |
.   3 |    6 |    3 |    9 ||     (3,   6) |  (15,   9) |       3 |
.   4 |    7 |    5 |   12 ||     (4,   8) |  (12,  16) |       4 |
.   5 |    1 |    7 |    8 ||    (10,   5) |   (5,  15) |       5 |
.   6 |    3 |   11 |   14 ||    (18,  12) |  (24,  30) |       6 |
.   7 |    1 |   29 |   30 ||    (70,   7) |   (7,  14) |       7 |
.   8 |    4 |   13 |   17 ||    (16,  24) |  (40,  32) |       8 |
.   9 |   10 |   10 |   20 ||     (9,  18) |  (36,  27) |       9 |
.  10 |   13 |   15 |   28 ||    (30,  20) |  (50,  70) |      10 |
.  11 |    1 |  122 |  123 ||   (660,  11) |  (11,  22) |      11 |
.  12 |    4 |   19 |   23 ||    (48,  36) |  (72,  60) |      12 |
.  13 |    1 |  200 |  201 ||  (1092,  13) |  (13,  26) |      13 |
.  14 |    1 |   31 |   32 ||    (14,  28) |  (28,  42) |      14 |
.  15 |    1 |   24 |   25 ||    (60,  45) |  (45,  75) |      15 |
.  16 |   21 |   18 |   39 ||    (32,  48) | (112,  64) |      16 |
.  17 |    1 |  299 |  300 ||  (2142,  17) |  (17,  34) |      17 |
.  18 |   24 |   22 |   46 ||    (54,  72) |  (90, 108) |      18 |
.  19 |    1 |  824 |  825 || (10260,  19) |  (19,  38) |      19 |
.  20 |   27 |   16 |   43 ||    (20,  40) |  (80, 100) |      20 |
.  21 |    1 |   33 |   34 ||    (42,  21) |  (21,  63) |      21 |
.  22 |    1 |  124 |  125 ||    (22,  44) |  (44,  66) |      22 |
.  23 |    1 |  945 |  946 || (12420,  23) |  (23,  46) |      23 |
.  24 |   10 |   41 |   51 ||    (96, 120) | (144, 168) |      24 |
.  25 |    1 |   26 |   27 ||    (75,  25) |  (25,  50) |      25 | .

Crossrefs

Cf. A256918, A257120, A257475, A257218.

Programs

  • Haskell
    a257478 n = a257475 n - a257120 n

Extensions

a(37)-a(72) from Hiroaki Yamanouchi, May 03 2015

A257218 Lexicographically earliest sequence of distinct positive integers such that gcd(a(n), a(n-1)) takes no value more than twice.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 10, 5, 15, 9, 18, 12, 16, 24, 30, 20, 40, 32, 48, 36, 27, 54, 72, 60, 45, 75, 25, 50, 70, 7, 14, 28, 42, 21, 63, 126, 84, 56, 112, 64, 96, 120, 80, 100, 150, 90, 108, 81, 162, 216, 144, 168, 140, 35, 105, 210, 180, 135, 225, 300
Offset: 1

Views

Author

Ivan Neretin, Apr 18 2015

Keywords

Comments

Presumably a(n) is a permutation of the positive integers.
Primes seem to occur in their natural order. 31 appears as a(7060). Primes p >= 37 are not found among the first 10000 terms.
Numbers n such that a(n)=n are 1, 2, 3, 12, 306, ...
A256918(n) = gcd(a(n), a(n+1)); gcd(a(A257120(n)), a(A257120(n)+1)) = gcd(a(A257475(n)), a(A257475(n)-1)) = n. - Reinhard Zumkeller, Apr 25 2015
For p prime: A257122(p)-1 = index of the smallest multiple of p: a(A257122(p)-1) mod p = 0 and a(m) mod p > 0 for m < A257122(p)-1. - Reinhard Zumkeller, Apr 26 2015

Examples

			After a(9)=15, the values 1, 2, 3, 4, 6, and 8 are already used, while 7 is forbidden because gcd(15,7)=1 and that value of GCD has already occurred twice, at (1,2) and (2,3). The minimal value which is neither used not forbidden is 9, so a(10)=9.

Links

Crossrefs

Other minimal sequences of distinct positive integers that match some condition imposed on a(n) and a(n-1):
A175498 (differences are unique),
A081145 (absolute differences are unique),
A235262 (bitwise XORs are unique),
A163252 (differ by one bit in binary),
A000027 (GCD=1),
A064413 (GCD>1),
A128280 (sum is a prime),
A034175 (sum is a square),
A175428 (sum is a cube),
A077220 (sum is a triangular number),
A073666 (product plus 1 is a prime),
A081943 (product minus 1 is a prime),
A091569 (product plus 1 is a square),
A100208 (sum of squares is a prime).
Cf. A004526.
Cf. A256918, A257120, A257475, A257478, A257122 (putative inverse).
Cf. also A281978.

Programs

  • Haskell
    import Data.List (delete); import Data.List.Ordered (member)
a257218 n = a257218_list !! (n-1)
a257218_list = 1 : f 1 [2..] a004526_list where
   f x zs cds = g zs where
     g (y:ys) | cd `member` cds = y : f y (delete y zs) (delete cd cds)
              | otherwise       = g ys
              where cd = gcd x y
-- Reinhard Zumkeller, Apr 24 2015
  • Mathematica
    a={1}; used=Array[0&,10000]; Do[i=1; While[MemberQ[a,i] || used[[l=GCD[a[[-1]],i]]]>=2, i++]; used[[l]]++; AppendTo[a,i], {n,2,100}]; a (* Ivan Neretin, Apr 18 2015 *)
Showing 1-4 of 4 results.

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