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 1-5 of 5 results.

A275716 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A273669(a(n)), a(2n+1) = A273664(a(n)).

Original entry on oeis.org

1, 2, 9, 3, 42, 17, 12, 4, 209, 115, 82, 41, 59, 26, 19, 5, 1042, 801, 572, 444, 409, 283, 202, 57, 292, 180, 129, 48, 92, 31, 22, 6, 5209, 5603, 4002, 4881, 2859, 3106, 2219, 733, 2042, 1977, 1412, 620, 1009, 395, 282, 97, 1459, 1258, 899, 525, 642, 334, 239, 74, 459, 213, 152, 63, 109, 40, 29, 7, 26042, 39217
Offset: 0

Views

Author

Antti Karttunen, Aug 06 2016

Keywords

Comments

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A273669(n) when the parent contains n, and each right hand child is obtained by applying A273664 to the parent's contents:
1
|
...................2...................
9 3
42......../ \........17 12......../ \........4
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
209 115 82 41 59 26 19 5
1042 801 572 444 409 283 202 57 292 180 129 48 92 31 22 6
etc.

Links

Crossrefs

Inverse: A275715.
Cf. A273664, A273669.
Related or similar permutations: A163511, A249824, A245612.

Formula

a(0) = 1, a(1) = 2, a(2n) = A273669(a(n)), a(2n+1) = A273664(a(n)).
As a composition of other permutations:
a(n) = A249824(A163511(n)).

A273669 Decimal representation ends with either 2 or 9.

Original entry on oeis.org

2, 9, 12, 19, 22, 29, 32, 39, 42, 49, 52, 59, 62, 69, 72, 79, 82, 89, 92, 99, 102, 109, 112, 119, 122, 129, 132, 139, 142, 149, 152, 159, 162, 169, 172, 179, 182, 189, 192, 199, 202, 209, 212, 219, 222, 229, 232, 239, 242, 249, 252, 259, 262, 269, 272, 279, 282, 289, 292, 299, 302, 309, 312, 319, 322, 329, 332, 339
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2016

Keywords

Comments

Natural numbers not in A273664.

Links

Crossrefs

Sequences A017293 and A017377 interleaved.
Cf. A000035, A001622, A007310, A084967, A126760, A249745, A249746.
Cf. also A273664, A249824, A275716.

Programs

  • Mathematica
    Select[Range@ 340, MemberQ[{2, 9}, Mod[#, 10]] &] (* or *)
Table[{10 n + 2, 10 n + 9}, {n, 0, 33}] // Flatten (* or *)
CoefficientList[Series[(-5/(1 - x) + (11 - x)/(-1 + x)^2 - 2/(1 + x))/2, {x, 0, 67}], x] (* Michael De Vlieger, Aug 07 2016 *)
  • Scheme
    (define (A273669 n) (+ (* 10 (/ (+ (- n 2) (if (odd? n) 1 0)) 2)) (if (odd? n) 2 9)))

Formula

a(n) = 10*(((n-2)+A000035(n))/2) + 2 [when n is odd], or + 9 [when n is even].
For n >= 5, a(n) = 2*a(n-2) - a(n-4).
a(n) = A126760(A084967(n)).
a(n) = A249746((3*A249745(n))-1).
Other identities. For all n >= 1:
A084967(n) = 5*A007310(n) = A007310(a(n)).
G.f.: x*(x^2+7*x+2)/((x+1)*(x-1)^2).
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((1+1/sqrt(5))/2)*phi^2*Pi/10 - log(phi)/(2*sqrt(5)) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, 15, 16, 59, 18, 41, 32, 20, 31, 39, 21, 23, 92, 40, 24, 49, 25, 27, 82, 48, 28, 209, 30, 45, 52, 33, 63, 62, 54, 34, 109, 35, 36, 129, 37, 38, 69, 43, 68, 142, 70, 57, 72, 115, 44, 79, 46, 85, 292, 47, 50, 89, 74, 73, 202, 51, 53, 159, 87, 55, 99, 107, 56, 152, 58, 97, 192, 60
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2014

Keywords

Comments

Permutation obtained from the odd bisection of A003961 (or from the odd bisection of A048673).

Examples

			a(5) = 9 because of the following. 2*A064216(5) = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(5) = 9.

Links

Crossrefs

Inverse: A249745.
Row 2 of A251722.
Cf. A003961, A007310, A048673, A084967, A126760, A249735, A064216, A249824.
Cf. also A273664, A273669.

Programs

  • Mathematica
    t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^6]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Map[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 #]] &, Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 87}]] (* Michael De Vlieger, Jul 25 2016, Version 10 *)
  • Scheme
    (define (A249746 n) (define (Ainv_of_A007310off0 n) (+ (* 2 (floor->exact (/ n 6))) (/ (- (modulo n 6) 1) 4))) (+ 1 (Ainv_of_A007310off0 (A003961 (+ n n -1)))))

Formula

a(n) = 1 + f(A003961(2n - 1)), where f(n) = 2*floor[n/6] + ((n mod 6)-1)/4. [Here 1 + f(A007310(n)) = n.]
a(n) = A126760(A249735(n)). - Antti Karttunen, Jul 25 2016
As a composition of related permutations:
a(n) = A249824(A064216(n)).
Other identities. For all n >= 1:
A249735(n) = A007310(a(n)).
a(3n-1) = A273669(a(n)) and a(A254049(n)) = A273664(a(n)). - Antti Karttunen, Aug 07 2016

A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

Original entry on oeis.org

1, 2, 3, 9, 4, 12, 5, 42, 17, 19, 6, 59, 7, 22, 26, 209, 8, 82, 10, 92, 31, 29, 11, 292, 41, 32, 115, 109, 13, 129, 14, 1042, 40, 39, 48, 409, 15, 49, 45, 459, 16, 152, 18, 142, 180, 52, 20, 1459, 57, 202, 54, 159, 21, 572, 63, 542, 68, 62, 23, 642, 24, 69, 213
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Examples

			a(4) = 9 because of the following. 2n = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(4) = 9.

Links

Crossrefs

Inverse: A249823.
Row 3 of A249822.
Cf. A003961, A048673, A078898, A084967, A243071, A246278, A249734, A249746, A249826, A250475, A275716.
Cf. also A273664, A273669.

Programs

  • Mathematica
    t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^4]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Table[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 n]], {n, 120}] (* Michael De Vlieger, Jul 25 2016, Version 10 *)
  • Scheme
    (define (A249824 n) (A078898 (A003961 (A003961 (* 2 n)))))

Formula

a(n) = A078898(A003961(A003961(2*n))) = A078898(A003961(A249734(n))).
a(n) = A078898(A246278(3,n)).
As a composition of other permutations:
a(n) = A249746(A048673(n)).
a(n) = A250475(A249826(n)).
a(n) = A275716(A243071(n)).
Other identities. For all n >= 1:
a(2n) = A273669(a(n)) and a(A003961(n)) = A273664(a(n)). -- Antti Karttunen, Aug 07 2016

A275715 Permutation of natural numbers: a(n) = A243071(A249823(n)).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 127, 2, 255, 511, 6, 1023, 2047, 4095, 8191, 5, 16383, 14, 32767, 65535, 30, 131071, 262143, 524287, 13, 1048575, 2097151, 62, 4194303, 29, 126, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 254, 61, 11, 4, 536870911, 1073741823, 125, 2147483647, 4294967295, 27, 510, 8589934591
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2016

Keywords

Comments

Note the indexing: the domain starts from 1, while the range includes also zero.

Links

Crossrefs

Inverse: A275716.
Related or similar permutations: A243071, A249823, A245611.
Cf. also A273664, A273669.

Programs

Formula

a(n) = A243071(A249823(n)).
Showing 1-5 of 5 results.

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