A034294 -id:A034294 - cp's OEIS frontend

A307498 Numbers k such that the digits of k in base 10 are a permutation of those of k in some other base.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 21, 23, 31, 41, 42, 43, 46, 51, 53, 61, 62, 63, 71, 73, 81, 82, 83, 84, 86, 91, 93, 158, 191, 196, 227, 261, 265, 283, 316, 370, 371, 441, 445, 511, 518, 551, 774, 782, 825, 834, 882, 910, 911, 912, 913, 914, 915, 916, 917, 918

Offset: 1

Jinyuan Wang, Aug 05 2019

Supersequence of A034294 and subsequence of A307498.
If the digits of k in base 10 is a permutation of m = (k in base b), 10^j < k < 10^(j+1), then 10^(j/(j+1)) < b < 10^((j+1)/j).
If k > 10, other base can only be 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 25, 26, 28, 37, 46, 55, 64, 73, 82.
The digits of k in base 10 is a permutation of k in base 82 iff k = 91.
The largest term is less than 10^25. See proof in A034294.

			13 in base 4 is 31, 227 in base 9 is 272.

Cf. A034294, A115920, A308493.

isok(k) = {my(v = vecsort(digits(k))); k < 10 || sum(j = 3, 82, vecsort(digits(k, j)) == v) > 1;}

A308493 Numbers k such that k in base 10 contains the same digits as k in some other base.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 20, 21, 23, 31, 41, 42, 43, 46, 51, 53, 61, 62, 63, 71, 73, 81, 82, 83, 84, 86, 91, 93, 100, 101, 102, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 131, 133, 141, 144, 151, 155, 158, 161, 166, 171, 177, 181

Offset: 1

Jinyuan Wang, Aug 05 2019

Supersequence of A034294 and A307498.
This sequence is infinite because 2*10^k is a term for any k >= 0.
Also 10^k is a term when k >= 0 and so too 10^k*(10^m - 1)/9 for any k > 0 and m >= 0. - Bruno Berselli, Aug 26 2019

			k = 113 is in the sequence because the set of digits of k {1, 3} equals the set of digits of (k in base 110) = 13.

Cf. A034294, A037415, A037422, A037428, A037433, A037437, A037440, A037442, A037443, A249899, A307498.

isok(k) = {my(j=Set(digits(k))); for(b=2, k+1, if((b!=10) && (Set(digits(k, b)) == j), return(1))); return(0);} \\ Michel Marcus, Aug 05 2019

A133377 Complete list of decimal numbers that when converted to hexadecimal produce the mirror image of the original number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 53, 371, 5141, 99481

Offset: 1

Daniel Mondot, Dec 21 2007

There are 14 numbers in all, including single-digit numbers.

			53 = 35_16; 371 = 173_16.

Subsequence of A133287. For n > 1, subsequence of A034294.

Select[Range[0,10^5],IntegerDigits[#]==Reverse[IntegerDigits[#,16]]&] (* James C. McMahon, Mar 17 2025 *)

isok(n) = digits(n, 10) == Vecrev(digits(n, 16)); \\ Michel Marcus, Oct 05 2019

A162572 Decimal numbers n which, when converted to a lower number base, show the reversed digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 13, 23, 46, 445, 2116, 15226, 313725, 1527465, 3454446, 2426472326, 3066511287, 102461381175475216

Offset: 1

Claudio Meller, Jul 06 2009

All entries lack the digit 9, because this digit is absent in the representations in bases 2 to 9.

The sequence is finite because for every n > 10^22 the representation of n in a base b < 10 has more digits than the representation of n in base 10. a(20) is the last term. - Giovanni Resta, Aug 13 2019

			13 is 31 in base 4. 23 is 32 in base 7. 46 is 64 in base 7.
801 is not a term since it is 1080 in base 9, so with an extra 0.

Cf. A034294, A090144. - R. J. Mathar, Jul 17 2009

isok(n) = {my(d = digits(n)); for (b=2, 9, my(rd = Vecrev(digits(n, b))); if ((#rd == #d) && fromdigits(rd) == n, return (b)););} \\ Michel Marcus, Aug 05 2019

Keyword:base, single-digit numbers and 313725 added by R. J. Mathar, Jul 17 2009
a(16)-a(17) from Michel Marcus, Aug 05 2019
a(18)-a(19) from Giovanni Resta, Aug 06 2019
a(20) from Giovanni Resta, Aug 13 2019

cp's OEIS Frontend

A307498 Numbers k such that the digits of k in base 10 are a permutation of those of k in some other base.

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A308493 Numbers k such that k in base 10 contains the same digits as k in some other base.

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A133377 Complete list of decimal numbers that when converted to hexadecimal produce the mirror image of the original number.

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A162572 Decimal numbers n which, when converted to a lower number base, show the reversed digits of n.

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