A262356 -id:A262356 - cp's OEIS frontend

A262358 Inverse permutation to A262356.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 13, 28, 30, 31, 35, 39, 43, 47, 51, 55, 15, 32, 38, 60, 61, 63, 65, 67, 69, 71, 17, 36, 42, 62, 77, 78, 80, 82, 84, 86, 19, 40, 46, 64, 79, 89, 90, 92, 94, 96, 21, 44, 50, 66, 81, 91, 99, 100

Offset: 1

Reinhard Zumkeller, Sep 19 2015

A262377 and A262377 give primes and where they occur: A262377(n)=a(A262378(n)).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (2958 terms from Reinhard Zumkeller)
Index entries for sequences that are permutations of the natural numbers

Cf. A262356, A262360 (fixed points), A262377 (primes), A262378.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a262358 = (+ 1) . fromJust . (`elemIndex` a262356_list)

terms = 100;
(* b = A262356 *) b[1] = 1; b[n_] := b[n] = Module[{s, k}, s = Rest[ IntegerDigits[b[n-1]]] //. {(0).., d__} :> {d}; For[k = 2, True, k++, If[FreeQ[Array[b, n-1], k], If[s == {0}, Return[k], If[IntegerDigits[ k][[1 ;; Length[s]]] == s, Return[k]]]]]];
Sort[Table[{b[n], n}, {n, 1, 2 terms}]][[1 ;; terms, 2]] (* Jean-François Alcover, Mar 12 2019 *)

A262363 Primes in A262356.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 101, 41, 103, 61, 71, 107, 29, 43, 53, 37, 73, 83, 109, 47, 59, 67, 79, 97, 89, 113, 401, 601, 701, 211, 503, 307, 311, 409, 509, 607, 709, 907, 809, 1009, 127, 131, 2003, 313, 4001, 137, 7001, 139, 9001, 1103, 3001, 1201

Offset: 1

Reinhard Zumkeller, Sep 20 2015

nonn

a(n) = A262356(A262371(n));

a permutation of the prime numbers, cf. A262377.

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

Cf. A262356, A010051, A262371, A262377.

a262363 n = a262363_list !! (n-1)
a262363_list = filter ((== 1) . a010051') a262356_list

A262390 Subsequence of terms starting with 1 in A262356.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 1000, 111, 112, 120, 113, 130, 114, 140, 115, 150, 116, 160, 117, 170, 118, 180, 119, 190, 1001, 121, 1100, 1002, 122, 123, 1003, 1101, 1010, 1004, 1005, 1006

Offset: 1

Reinhard Zumkeller, Sep 21 2015

A000030(a(n)) = 1;

A262356(A262393(n)) = a(n).

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

Cf. A262356, A000030, A262393, A131835.

a262390 n = a262390_list !! (n-1)
a262390_list = filter ((== 1) . a000030) a262356_list

Typo in name corrected by Andrey Zabolotskiy, Sep 22 2017

A262360 Fixed points of A262356 and of A262358.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 42, 4410, 3060996

Offset: 1

Reinhard Zumkeller, Sep 19 2015

A261466(a(n)) = a(n).

Table of n, a(n) for n = 1..15

Cf. A262356, A262358.

a262360 n = a262360_list !! (n-1)
a262360_list = [x | x <- [1..], a262356 x == x]

a(15) from Alois P. Heinz, Sep 20 2015

A262371 Positions of prime numbers in A262356.

Original entry on oeis.org

2, 3, 5, 7, 11, 14, 22, 26, 31, 32, 33, 36, 37, 44, 48, 53, 55, 62, 64, 67, 68, 70, 74, 82, 96, 100, 110, 111, 114, 127, 131, 141, 146, 163, 176, 179, 187, 200, 211, 216, 227, 228, 232, 235, 251, 260, 267, 268, 274, 281, 283, 287, 289, 292, 294, 299, 314

Offset: 1

Reinhard Zumkeller, Sep 20 2015

nonn

In other words, the n-th prime appears in A262356 at position a(n). - N. J. A. Sloane, Sep 29 2015

A262363(n) = A262356(a(n)).

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

Cf. A262356, A010051, A262363.

a262371 n = a262371_list !! (n-1)
a262371_list = filter ((== 1) . a010051' . a262356) [1..]

A262393 Positions of numbers starting with 1 in A262356.

Original entry on oeis.org

1, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 29, 33, 34, 37, 41, 45, 49, 53, 57, 74, 119, 120, 121, 122, 123, 127, 128, 132, 133, 137, 138, 142, 143, 147, 148, 152, 153, 157, 158, 161, 162, 164, 165, 167, 170, 186, 188, 189, 190, 203, 214, 223, 230, 235, 238

Offset: 1

Reinhard Zumkeller, Sep 21 2015

A262356(a(n)) = A262390(n).

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

Cf. A000030, A262356, A262390.

a262393 n = a262393_list !! (n-1)
a262393_list = filter ((== 1) . a000030 . a262356) [1..]

A262282 a(1)=11. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

Original entry on oeis.org

11, 13, 3, 2, 5, 7, 17, 71, 19, 97, 73, 31, 101, 103, 37, 79, 907, 701, 107, 709, 911, 113, 131, 311, 1103, 1031, 313, 137, 373, 733, 331, 317, 173, 739, 397, 971, 719, 191, 919, 193, 937, 379, 797, 977, 773, 7307, 307, 727, 271, 7103, 1033, 337, 3701, 7013

Offset: 1

Franklin T. Adams-Watters and N. J. A. Sloane, Sep 18 2015

If a(n-1) has a single digit then a(n) is simply the smallest missing prime.
Leading zeros in s are ignored.
The b-file suggests that there are infinitely many primes that do not appear in the sequence. However, there is no proof at present that any particular prime (23, say) never appears.
Alois P. Heinz points out that this sequence and A262283 eventually merge (see the latter entry for details). - N. J. A. Sloane, Sep 19 2015
A variant without the prime number condition: A262356. - Reinhard Zumkeller, Sep 19 2015

			a(1)=11, so s=1, a(2) is smallest missing prime that starts with 1, so a(2)=13. Then s=3, so a(3)=3. Then s is the empty string, so a(4)=2, and so on.

Alois P. Heinz, Table of n, a(n) for n = 1..705

Suggested by A089755. Cf. A262283.

Cf. A262356.

import Data.List (isPrefixOf, delete)
a262282 n = a262282_list !! (n-1)
a262282_list = 11 : f "1" (map show (delete 11 a000040_list)) where
   f xs pss = (read ys :: Integer) :
              f (dropWhile (== '0') ys') (delete ys pss)
              where ys@(_:ys') = head $ filter (isPrefixOf xs) pss
-- Reinhard Zumkeller, Sep 19 2015

More terms from Alois P. Heinz, Sep 18 2015

A262374 a(1) = 1; for n > 1, let s denote the binary representation of a(n-1) with the first bit omitted. Then a(n) is the smallest number not yet present whose binary representation starts with s, omitting leading zeros.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 12, 9, 10, 11, 13, 20, 16, 14, 24, 17, 15, 28, 25, 18, 19, 26, 21, 22, 27, 23, 29, 52, 40, 32, 30, 56, 48, 33, 31, 60, 57, 50, 36, 34, 35, 49, 68, 37, 41, 38, 51, 39, 58, 53, 42, 43, 44, 96, 64, 45, 54, 88, 97, 66, 46, 59, 55, 47, 61

Offset: 1

Allan C. Wechsler, Sep 20 2015

It seems clear that every number will appear. It would be nice to have a formal proof. - N. J. A. Sloane, Sep 20 2015

			: 1                                      ...  1
:  10                                    ...  2
:    11                                  ...  3
:     100                                ...  4
:        101                             ...  5
:          110                           ...  6
:           1000                         ...  8
:               111                      ...  7
:                1100                    ... 12
:                 1001                   ...  9
:                    1010                ... 10
:                      1011              ... 11
:                        1101            ... 13
:                         10100          ... 20
:                           10000        ... 16
:                                1110    ... 14
:                                 11000  ... 24
:                                  10001 ... 17

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Binary counterpart of A262356.
A262381 gives the binary representations.
Cf. A262388.

cp's OEIS Frontend

A262358 Inverse permutation to A262356.

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A262363 Primes in A262356.

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A262390 Subsequence of terms starting with 1 in A262356.

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A262360 Fixed points of A262356 and of A262358.

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A262371 Positions of prime numbers in A262356.

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A262393 Positions of numbers starting with 1 in A262356.

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A262282 a(1)=11. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

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A262374 a(1) = 1; for n > 1, let s denote the binary representation of a(n-1) with the first bit omitted. Then a(n) is the smallest number not yet present whose binary representation starts with s, omitting leading zeros.

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