A228407 -id:A228407 - cp's OEIS frontend

A231930 Working in base 8: a(0)=0, thereafter a(n+1) is the smallest number not already in the sequence such that the bits of a(n) and a(n+1) together can be rearranged to form a palindrome.

0, 11, 1, 10, 100, 12, 2, 20, 101, 22, 3, 13, 31, 103, 30, 110, 33, 4, 14, 41, 104, 40, 114, 24, 42, 112, 21, 102, 120, 201, 210, 1000, 105, 15, 5, 25, 52, 115, 35, 53, 113, 23, 32, 121, 26, 6, 16, 61, 106, 60, 116, 36, 63, 131, 34, 43, 134, 143, 314, 341, 413, 431, 1003

Offset: 0

N. J. A. Sloane and Robert G. Wilson v, Nov 15 2013

This is a permutation of the nonnegative integers in base 8 - see the Comments in A228407 for the proof.

Cf. A230891, A231920, A231922, A231924, A231926, A231928, A231932, A228407, A231931.

a[0] = 0; a[n_] := a[n] = Block[{k = 1, idm = IntegerDigits[ a[n - 1], 8], t = a@# & /@ Range[n - 1]}, Label[ start]; While[ MemberQ[t, k], k++];  While[ Select[ Permutations[ Join[ idm, IntegerDigits[k, 8]]], #[[1]] != 0 && # == Reverse@# &] == {}, k++; Goto[ start]]; k]; s = Array[a, 60, 0]; FromDigits@# & /@ IntegerDigits[s, 8]

A231931 Terms of A231930 written in base 10: the binary expansions of a(n) and a(n+1) taken together can be rearranged to form a palindrome.

Original entry on oeis.org

0, 9, 1, 8, 64, 10, 2, 16, 65, 18, 3, 11, 25, 67, 24, 72, 27, 4, 12, 33, 68, 32, 76, 20, 34, 74, 17, 66, 80, 129, 136, 512, 69, 13, 5, 21, 42, 77, 29, 43, 75, 19, 26, 81, 22, 6, 14, 49, 70, 48, 78, 30, 51, 89, 28, 35, 92, 99, 204, 225, 267, 281, 515, 73, 15, 7, 23, 58

Offset: 0

N. J. A. Sloane and Robert G. Wilson v, Nov 15 2013

See A231930 for precise definition.

This is a permutation of the nonnegative integers - see the Comments in A228407 for the proof.

Cf. A230892, A231921, A231923, A231925, A231927, A231929, A231933, A228407, A231930.

a[0] = 0; a[n_] := a[n] = Block[{k = 1, idm = IntegerDigits[ a[n - 1], 8], t = a@# & /@ Range[n - 1]}, Label[ start]; While[ MemberQ[t, k], k++];  While[ Select[ Permutations[ Join[ idm, IntegerDigits[k, 8]]], #[[1]] != 0 && # == Reverse@# &] == {}, k++; Goto[ start]]; k]; s = Array[a, 60, 0]

A231932 Working in base 9: a(0)=0, thereafter a(n+1) is the smallest number not already in the sequence such that the bits of a(n) and a(n+1) together can be rearranged to form a palindrome.

Original entry on oeis.org

0, 11, 1, 10, 100, 12, 2, 20, 101, 22, 3, 13, 31, 103, 30, 110, 33, 4, 14, 41, 104, 40, 114, 24, 42, 112, 21, 102, 120, 201, 210, 1000, 105, 15, 5, 25, 52, 115, 35, 53, 113, 23, 32, 121, 26, 6, 16, 61, 106, 60, 116, 36, 63, 131, 34, 43, 134, 143, 314, 341, 413, 431, 1003

Offset: 0

N. J. A. Sloane and Robert G. Wilson v, Nov 15 2013

This is a permutation of the nonnegative integers in base 9 - see the Comments in A228407 for the proof.

Cf. A230891, A231920, A231922, A231924, A231926, A231928, A231930, A228407, A231933.

a[0] = 0; a[n_] := a[n] = Block[{k = 1, idm = IntegerDigits[ a[n - 1], 9], t = a@# & /@ Range[n - 1]}, Label[ start]; While[ MemberQ[t, k], k++];  While[ Select[ Permutations[ Join[ idm, IntegerDigits[k, 9]]], #[[1]] != 0 && # == Reverse@# &] == {}, k++; Goto[ start]]; k]; s = Array[a, 60, 0]; FromDigits@# & /@ IntegerDigits[s, 9]

A228410 The digits of a(n) and a(n+1) together can be reordered to form a palindrome; lexicographically least injective sequence of positive integers with this property.

Original entry on oeis.org

1, 10, 100, 11, 2, 12, 21, 102, 20, 101, 22, 3, 13, 31, 103, 30, 110, 33, 4, 14, 41, 104, 40, 114, 24, 42, 112, 23, 32, 113, 34, 43, 131, 35, 5, 15, 51, 105, 50, 115, 25, 52, 121, 26, 6, 16, 61, 106, 60, 116, 36, 63, 136, 163, 316, 361, 613, 631, 1003, 111, 17, 7, 27, 72, 117, 37, 73, 137, 71, 107

Offset: 1

M. F. Hasler, Nov 09 2013

For each n=1,2,3..., choose the smallest positive integer a(n) not occurring earlier such that the digits of a(n) and the preceding term (none for n=1) taken together can form a palindrome, when suitably reordered.
This is a variant of the original version, proposed by E. Angelini, based on nonnegative integers (cf. A228407). The two sequences start with only a few terms differing and large segments in common, and one might have expected them to join a common orbit quite early, but they rather diverge more and more.
It is conjectured that the sequence is a permutation of the positive integers, i.e., that all numbers will eventually occur. To test this conjecture, one can consider the indices n at which occur the numbers equal to the smallest integer not yet used. If the conjecture is true, this is equivalent to a(m)>a(n) for all m>n; if not, then this list ends at the first missing number. These [n,a(n)] are: [1, 1], [5, 2], [12, 3], [19, 4], [35, 5], [45, 6], [62, 7], [78, 8], [88, 9], [89, 29], [92, 39], [118, 44], [149, 45], [187, 46], [314, 47], [432, 49], [477, 59], [506, 67], [507, 76], [521, 78], [531, 79], [572, 89], [573, 98], [574, 198], [954, 211][955, 222], [956, 233], [1602, 234], [1616, 235], [1623, 237], [1924, 238], [1959, 239], [2508, 258], [2515, 278], [2536, 279], [4046, 289], [4047, 298], [4053, 489], [4054, 498], ...
Sequence A228412 is an "arithmetic" variant, where instead of the union of the digits, the sum of terms is considered. Sequence A062932 is a further variant where injectivity is replaced by monotonicity.
Sequences A231433 and A231442 are variants where "palindrome" is replaced with "prime".

Lars Blomberg, Table of n, a(n) for n = 1..10000
E. Angelini, Re: Two make a palindrome, SeqFan list, Nov 09 2013

Cf. A228407, A231442, A231433.

PARI
```
{u=0; a=1; for(n=1,99, u+=1<
				
```

from collections import Counter
A228410_list, l, s, b = [1], Counter('1'), 2, set()
for _ in range(10**2):
    i = s
    while True:
        if i not in b:
            li, o = Counter(str(i)), 0
            for d in (l+li).values():
                if d % 2:
                    if o > 0:
                        break
                    o += 1
            else:
                A228410_list.append(i)
                l = li
                b.add(i)
                while s in b:
                    b.remove(s)
                    s += 1
                break
        i += 1 # Chai Wah Wu, Dec 14 2014

A231433 The digits of a(n) and a(n+1) taken together are the digits of a prime; least permutation of the nonnegative integers with this property.

Original entry on oeis.org

0, 11, 2, 3, 1, 4, 7, 6, 10, 9, 5, 12, 8, 18, 13, 15, 14, 17, 20, 23, 21, 16, 19, 22, 30, 25, 27, 26, 29, 24, 31, 28, 33, 32, 35, 36, 34, 37, 39, 38, 41, 42, 43, 45, 47, 44, 51, 40, 49, 46, 57, 50, 53, 48, 59, 56, 63, 52, 61, 54, 67, 55, 69, 58, 70, 60, 71, 62, 72

Offset: 0

Eric Angelini and M. F. Hasler, Nov 09 2013

The offset is zero to have a permutation.
Sequence A128280 is an "arithmetic" analog, where instead of concatenation of digits, the terms are added.
Sequences A228407 and A228410 are the variants where "prime" is replaced by "palindrome".

			Start with a(0)=0. The least prime having this digit is 101, so a(1)=11. Since 0 cannot be used any more and 111 is not a prime, the least digit that can be added to get the digits of some prime (namely 211) is a(2)=2, then a(3)=3 yields 23, etc.
See also the link to Angelini's post.

Lars Blomberg, Table of n, a(n) for n = 0..9999
E. Angelini, Two make a prime, Nov 09 2013

{a=u=0;for(n=1,99,u+=1<"0"&&(a=k)&&next(3))))}

A231889 Base 2 analog of A231880 (written in base 2).

Original entry on oeis.org

0, 10, 101, 100, 1, 1001, 111, 1010, 1000, 11, 10000, 110, 1011, 10001, 1101, 10010, 1110, 10011, 1100, 10101, 10100, 1111, 11000, 10110, 11001, 11010, 11100, 100001, 10111, 100000, 11011, 100010, 11101

Offset: 0

N. J. A. Sloane, Nov 17 2013

A231889 (and A231890) eventually merges with A231891 (and A231892). For n>30, a(n) = A231891(n-1). - Hans Havermann, Nov 18 2013

Hans Havermann, Table of n, a(n) for n = 0..1000

Cf. A228407, A231880, A231881, A231889, A231890, A231891, A231892.

More terms from Hans Havermann, Nov 18 2013

A231890 Terms of A231889 (base-2 analog of A231880) written in base 10.

Original entry on oeis.org

0, 2, 5, 4, 1, 9, 7, 10, 8, 3, 16, 6, 11, 17, 13, 18, 14, 19, 12, 21, 20, 15, 24, 22, 25, 26, 28, 33, 23, 32, 27, 34, 29, 35, 30, 36, 31, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 47, 49, 50

Offset: 0

N. J. A. Sloane, Nov 17 2013

A231889 (and A231890) eventually merges with A231891 (and A231892). For n > 30, a(n) = A231892(n-1). - Hans Havermann, Nov 18 2013

Hans Havermann, Table of n, a(n) for n = 0..1000

Cf. A228407, A231880, A231881, A231889, A231890, A231891, A231892.

More terms from Hans Havermann, Nov 18 2013

A231891 Base-2 analog of A231881 (written in base 2).

Original entry on oeis.org

1, 100, 11, 1000, 10, 101, 1011, 110, 1101, 10001, 1110, 10010, 1111, 10000, 111, 1001, 1010, 1100, 10011, 10100, 10101, 10110, 11000, 10111, 100000, 11011, 100001, 11001, 11010, 11100, 100010, 11101, 100011

Offset: 0

N. J. A. Sloane, Nov 17 2013

A231889 (and A231890) eventually merges with A231891 (and A231892). For n > 29, a(n) = A231889(n+1). - Hans Havermann, Nov 18 2013

Hans Havermann, Table of n, a(n) for n = 0..1000

Cf. A228407, A231880, A231881, A231889-A231892.

More terms from Hans Havermann, Nov 18 2013

A231892 Terms of A231891 (base-2 analog of A231881) written in base 10.

Original entry on oeis.org

1, 4, 3, 8, 2, 5, 11, 6, 13, 17, 14, 18, 15, 16, 7, 9, 10, 12, 19, 20, 21, 22, 24, 23, 32, 27, 33, 25, 26, 28, 34, 29, 35, 30, 36, 31, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 47, 49, 50, 51

Offset: 0

N. J. A. Sloane, Nov 17 2013

A231889 (and A231890) eventually merges with A231891 (and A231892). For n > 29, a(n) = A231890(n+1). - Hans Havermann, Nov 18 2013

Hans Havermann, Table of n, a(n) for n = 0..1000

Cf. A228407, A231880, A231881, A231889, A231890, A231891, A231892.

For n > 29, a(n) = A231890(n+1). - Hans Havermann, Nov 18 2013

More terms from Hans Havermann, Nov 18 2013

A308719 Lexicographically earliest sequence of distinct terms such that the digits of two contiguous terms sum up to a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 7, 11, 9, 20, 12, 8, 21, 13, 14, 15, 23, 22, 16, 31, 25, 40, 30, 17, 59, 26, 68, 35, 77, 44, 86, 53, 95, 62, 100, 19, 39, 28, 48, 37, 57, 46, 66, 55, 75, 64, 84, 73, 93, 82, 129, 91, 138, 109, 147, 118, 156, 127, 165, 136, 174, 145, 183, 154, 192, 163, 219, 172, 228, 181, 237, 190

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jun 19 2019

This sequence is not a permutation of the integers > 0 as integers with digitsum 11, or 22, or 33, for instance, will not show.

			The sequence starts with 1,2,3,4,5,6,10,7,11,9,... and we see indeed that the digits of:
{a(1); a(2)} have sum 1 + 2 = 3 (palindrome);
{a(2); a(3)} have sum 2 + 3 = 5 (palindrome);
{a(3); a(4)} have sum 3 + 4 = 7 (palindrome);
{a(4); a(5)} have sum 4 + 5 = 9 (palindrome);
{a(5); a(6)} have sum 5 + 6 = 11 (palindrome);
{a(6); a(7)} have sum 6 + 1 + 0 = 7 (palindrome);
{a(7); a(8)} have sum 1 + 0 + 7 = 8 (palindrome);
{a(8); a(9)} have sum 7 + 1 + 1 = 9 (palindrome);
{a(9); a(10)} have sum 1 + 1 + 9 = 11 (palindrome);
etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10001

Cf. A308727 with squares instead of palindromes and A308728 with primes.

Cf. A228407.

a[1]=1; a[n_]:=a[n]=(k=1;While[MemberQ[Array[a,n-1],k]|| !PalindromeQ@Total[Join[IntegerDigits@a[n-1],IntegerDigits@k]], k++];k)
Array[a,68] (* Giorgos Kalogeropoulos, Jul 14 2023 *)

cp's OEIS Frontend

A231930 Working in base 8: a(0)=0, thereafter a(n+1) is the smallest number not already in the sequence such that the bits of a(n) and a(n+1) together can be rearranged to form a palindrome.

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A231931 Terms of A231930 written in base 10: the binary expansions of a(n) and a(n+1) taken together can be rearranged to form a palindrome.

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A231932 Working in base 9: a(0)=0, thereafter a(n+1) is the smallest number not already in the sequence such that the bits of a(n) and a(n+1) together can be rearranged to form a palindrome.

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A228410 The digits of a(n) and a(n+1) together can be reordered to form a palindrome; lexicographically least injective sequence of positive integers with this property.

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A231433 The digits of a(n) and a(n+1) taken together are the digits of a prime; least permutation of the nonnegative integers with this property.

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A231889 Base 2 analog of A231880 (written in base 2).

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A231890 Terms of A231889 (base-2 analog of A231880) written in base 10.

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A231891 Base-2 analog of A231881 (written in base 2).

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A231892 Terms of A231891 (base-2 analog of A231881) written in base 10.

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A308719 Lexicographically earliest sequence of distinct terms such that the digits of two contiguous terms sum up to a palindrome.

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