A056965 -id:A056965 - cp's OEIS frontend

A278328 Numbers n such that abs(n - rev(n)) is a square, where rev(n) is the decimal reverse of n (A004086).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 21, 22, 23, 26, 32, 33, 34, 37, 40, 43, 44, 45, 48, 51, 54, 55, 56, 59, 62, 65, 66, 67, 73, 76, 77, 78, 84, 87, 88, 89, 90, 95, 98, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262

Offset: 1

Jonathan Frech, Nov 18 2016

All palindromes are in this sequence, hence it is infinite.

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

A002113 is a subsequence.

Cf. A000290, A004086, A016766, A056965.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, -1, a(n-1)) while not issqr(abs(k-(s->
       parse(cat(s[-i]$i=1..length(s))))(""||k))) do od: k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 18 2016

Select[Range@ 262, IntegerQ@ Sqrt@ Abs[# - FromDigits@ Reverse@ IntegerDigits@ #] &] (* Michael De Vlieger, Nov 18 2016 *)

is(n) = issquare(abs(n - eval(concat(Vecrev(Str(n)))))) \\ Felix Fröhlich, Nov 18 2016

is(n, {b=10}) = issquare(abs(n - subst(Polrev(digits(n, b),'x),'x,b))); \\ Gheorghe Coserea, Nov 27 2016

import math
n = 0
while True:
    if math.sqrt(abs(n-int(str(n)[::-1])))%1 == 0:
        print(n)
    n += 1 # Jonathan Frech, Nov 18 2016

A335978 Numbers m of the form abs(k - reverse(k)) for at least one k.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 180, 189, 198, 270, 279, 297, 360, 369, 396, 450, 459, 495, 540, 549, 594, 630, 639, 693, 720, 729, 792, 810, 819, 891, 900, 909, 990, 999, 1089, 1179, 1188, 1269, 1278, 1359, 1368, 1449, 1458, 1539, 1548, 1629, 1638, 1719, 1728, 1800, 1809, 1818, 1890, 1908, 1980, 1989, 1998, 2079

Offset: 1

Michael Greaney, Jul 03 2020

All terms are divisible by 9.
Let f(k) = k - reverse(k). Then f(reverse(k)) = -f(k), since f(reverse(k)) = reverse(k) - reverse(reverse(k)) = reverse(k) - k = - (k - reverse(k)) = -f(k).
Iteration of the function f(k) = k - reverse(k) leads to A072140, A072141, A072142, and A072143.

Michael P. Greaney, Remarkable Reversible Numbers, +plus magazine, (September 29, 2015).

Cf. A056965, A067030, A072140, A072141, A072142, A072143.

Dividing by 9 gives A334145.

A338827 For any number with decimal representation (d(1), d(2), ..., d(k)), the decimal representation of a(n) is (abs(d(1)-d(k)), abs(d(2)-d(k-1)), ..., abs(d(k)-d(1))).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 11, 22, 33, 44, 55, 66, 77, 88, 22, 11, 0, 11, 22, 33, 44, 55, 66, 77, 33, 22, 11, 0, 11, 22, 33, 44, 55, 66, 44, 33, 22, 11, 0, 11, 22, 33, 44, 55, 55, 44, 33, 22, 11, 0, 11, 22, 33, 44, 66, 55, 44, 33, 22, 11, 0, 11, 22

Offset: 0

Rémy Sigrist, Nov 11 2020

Leading zeros are ignored.

All terms belong to A061917.

			For n = 1021:
- abs(1-1) = 0,
- abs(0-2) = 2,
- abs(2-0) = 2,
- abs(1-1) = 0,
- so a(1021) = 220.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A002113, A004086, A056965, A061917, A175919 (binary analog), A330240, A338828 (ternary analog).

a:= n-> (l-> (h-> add(h[j]*10^(j-1), j=1..nops(h)))([seq(
    abs(l[i]-l[-i]), i=1..nops(l))]))(convert(n, base, 10)):
seq(a(n), n=0..70);  # Alois P. Heinz, Nov 12 2020

a(n, base=10) = my (d=digits(n, base)); fromdigits(abs(d-Vecrev(d)), base)

a(n) = 0 iff n is a palindrome (A002113).

a(n) = A330240(n, A004086(n)).

A349240 a(n) = n - (reversal of digits in the Zeckendorf representation of n).

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 0, 3, 7, 0, 4, 7, 0, 12, 0, 6, 10, -2, 14, 2, 8, 20, 0, 9, 15, -5, 20, 0, 9, 25, 5, 14, 20, 0, 33, 0, 14, 23, -10, 30, -3, 11, 36, 3, 17, 26, -7, 43, 10, 24, 33, 0, 40, 7, 21, 54, 0, 22, 36, -18, 46, -8, 14, 54, 0, 22, 36, -18, 62, 8, 30, 44

Offset: 0

Kevin Ryde, Nov 11 2021

Kevin Ryde, Table of n, a(n) for n = 0..10000
Kevin Ryde, PARI/GP Code

Cf. A189920 (Zeckendorf digits), A349238 (reverse), A349239 (reverse and add).
Cf. A094202 (indices of 0's).
Other bases: A055945 (binary), A056965 (decimal).

PARI
```
\\ See links.
```

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
print(a - a, end = ", ")
while n < 71:
    n += 1
    print(n - RevFibToNum(NumToFib(n)), end = ", ") # A.H.M. Smeets, Nov 14 2021

a(n) = n - A349238(n).

a(n) = 2*n - A349239(n).

A055949 n - reversal of base 4 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 3, 0, -3, -6, 6, 3, 0, -3, 9, 6, 3, 0, 15, 0, -15, -30, 15, 0, -15, -30, 15, 0, -15, -30, 15, 0, -15, -30, 30, 15, 0, -15, 30, 15, 0, -15, 30, 15, 0, -15, 30, 15, 0, -15, 45, 30, 15, 0, 45, 30, 15, 0, 45, 30, 15, 0, 45, 30, 15, 0, 63, 0, -63, -126, 51, -12, -75, -138, 39, -24, -87, -150, 27, -36, -99, -162, 75, 12, -51

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 3.

			For n = 6, the reversal of base 4 digits of n (written in base 10) is 9. So, a(6) = 6 - 9 = -3. - _Indranil Ghosh_, Feb 01 2017

Indranil Ghosh, Table of n, a(n) for n = 0..20000

Cf. A030103, A056965.

Table[n-FromDigits[Reverse[IntegerDigits[n,4]],4],{n,0,90}] (* Harvey P. Dale, Aug 22 2011 *)

a(n) = n - A030103(n).

A055951 n - reversal of base 5 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, -4, -8, -12, 8, 4, 0, -4, -8, 12, 8, 4, 0, -4, 16, 12, 8, 4, 0, 24, 0, -24, -48, -72, 24, 0, -24, -48, -72, 24, 0, -24, -48, -72, 24, 0, -24, -48, -72, 24, 0, -24, -48, -72, 48, 24, 0, -24, -48, 48, 24, 0, -24, -48, 48, 24, 0, -24, -48, 48, 24, 0, -24, -48, 48, 24, 0, -24, -48, 72, 48, 24, 0, -24, 72, 48, 24, 0

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 4.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A030104, A056965.

Table[n - IntegerReverse[n, 5], {n, 0, 100}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = n - A030104(n).

A055953 n - reversal of base 6 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 5, 0, -5, -10, -15, -20, 10, 5, 0, -5, -10, -15, 15, 10, 5, 0, -5, -10, 20, 15, 10, 5, 0, -5, 25, 20, 15, 10, 5, 0, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 70, 35, 0, -35, -70

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 5.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A030105, A056965.

Table[n - IntegerReverse[n, 6], {n, 0, 100}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = n - A030105(n).

A055961 a(n) = n - (reversal of base-11 digits of n) (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, -10, -20, -30, -40, -50, -60, -70, -80, -90, 20, 10, 0, -10, -20, -30, -40, -50, -60, -70, -80, 30, 20, 10, 0, -10, -20, -30, -40, -50, -60, -70, 40, 30, 20, 10, 0, -10, -20, -30, -40, -50, -60, 50, 40, 30, 20, 10, 0, -10, -20, -30, -40, -50, 60, 50, 40, 30, 20, 10, 0, -10, -20, -30, -40

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 10.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A056960 (base-11 reversal), A056965 (n minus decimal reversal).

f:= proc(n) local L,t;
  L:= convert(n,base,11);
  n - add(L[-i]*11^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Apr 20 2021

Table[n-FromDigits[Reverse[IntegerDigits[n,11]],11],{n,0,80}] (* Harvey P. Dale, Feb 21 2023 *)

a(n) = n - fromdigits(Vecrev(digits(n, 11)), 11); \\ Michel Marcus, Apr 22 2021

a(n) = n - A056960(n).

A055963 n - reversal of base 12 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, -11, -22, -33, -44, -55, -66, -77, -88, -99, -110, 22, 11, 0, -11, -22, -33, -44, -55, -66, -77, -88, -99, 33, 22, 11, 0, -11, -22, -33, -44, -55, -66, -77, -88, 44, 33, 22, 11, 0, -11, -22, -33, -44, -55, -66, -77, 55, 44, 33, 22, 11, 0, -11, -22, -33, -44, -55, -66, 66, 55, 44

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 11.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A056961, A056965.

Table[n - IntegerReverse[n, 12], {n, 0, 100}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = n - A056961(n).

A383887 Smallest non-palindromic number that is congruent to its reverse mod n.

Original entry on oeis.org

10, 13, 10, 15, 16, 13, 18, 19, 10, 1011, 100, 15, 1017, 1027, 16, 1025, 1039, 13, 1048, 1021, 18, 103, 1026, 19, 1026, 1017, 14, 1033, 1013, 1011, 1068, 1049, 100, 1039, 1046, 15, 1000, 1055, 1017, 1041, 1066, 1027, 1048, 105, 16, 1077, 1032, 1025, 1014, 1051, 1039, 1017, 1103, 17, 106, 1065

Offset: 1

Erick B. Wong, May 29 2025, at the suggestion of Lanny Wong

Comparable to A070837, but such a number provably exists: if n is coprime to 10 then take 10^k where k is the order of 10 mod n; else take a>b>0 such that n divides 10^a - 10^b, then the number 10^(a+b) + 10^b + 1 works.

The n-th term in this sequence is equal to the first nonzero term of A070837 that occurs at an index divisible by n/gcd(n,9).

			For n=6, a(6)=13 is congruent to 31 mod 6.
For n=10, note that any number of 3 or fewer digits is necessarily a palindrome if the first digit equals the last, and 1011 is the first 4-digit non-palindrome.

Erick B. Wong, Table of n, a(n) for n = 1..10000

Cf. A004086, A056965, A070837.

seq[len_] := Module[{s = Table[0, {len}], c = 0, k = 9, d}, While[c < len, k++; krev = IntegerReverse[k]; If[k != krev, d = Select[Divisors[Abs[k - krev]], # <= len &]; Do[If[s[[d[[i]]]] == 0, s[[d[[i]]]] = k; c++], {i, 1, Length[d]}]]]; s]; seq[60] (* Amiram Eldar, May 31 2025 *)

a(n) = my(k=1, d=digits(k), rd=Vecrev(d)); while(!((d != rd) && Mod(fromdigits(rd), n) == k), k++; d=digits(k); rd=Vecrev(d)); k; \\ Michel Marcus, May 30 2025

from functools import partial
from itertools import count
def accept(mod, k):
    r = int(str(k)[::-1])
    return r != k and (r-k) % mod == 0
def a(n):
    return next(filter(partial(accept, n), count(10)))
print([a(n) for n in range(1,57)])

cp's OEIS Frontend

A278328 Numbers n such that abs(n - rev(n)) is a square, where rev(n) is the decimal reverse of n (A004086).

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A335978 Numbers m of the form abs(k - reverse(k)) for at least one k.

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A338827 For any number with decimal representation (d(1), d(2), ..., d(k)), the decimal representation of a(n) is (abs(d(1)-d(k)), abs(d(2)-d(k-1)), ..., abs(d(k)-d(1))).

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A349240 a(n) = n - (reversal of digits in the Zeckendorf representation of n).

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A055949 n - reversal of base 4 digits of n (written in base 10).

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A055951 n - reversal of base 5 digits of n (written in base 10).

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A055953 n - reversal of base 6 digits of n (written in base 10).

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A055961 a(n) = n - (reversal of base-11 digits of n) (written in base 10).

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