A029742 -id:A029742 - cp's OEIS frontend

A118959 Non-palindromic numbers which are divisible by their reversal.

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 200, 220, 300, 330, 400, 440, 500, 510, 540, 550, 600, 660, 700, 770, 800, 810, 880, 900, 990, 1000, 1010, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2000, 2020, 2100, 2120, 2200, 2220, 2320, 2420

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 25 2006

A004086(a(n)) = A027751(a(n),k) for some k: 1 <= k < A001221(n); A031877 contains all terms not ending with zero. - Reinhard Zumkeller, Jul 15 2013

			510 is in the sequence because 510 is a non-palindromic number divisible by its reversal 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A029742; A031877 is a subsequence.

a118959 n = a118959_list !! (n-1)
a118959_list = filter
   (\x -> let x' = a004086 x in x' /= x && x `mod` x' == 0) [1..]
-- Reinhard Zumkeller, Jul 15 2013

Select[Range[2420],!PalindromeQ[#]&&Divisible[#,IntegerReverse[#]]&] (* James C. McMahon, Sep 13 2024 *)

A319388 Non-palindromic squares.

Original entry on oeis.org

16, 25, 36, 49, 64, 81, 100, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 529, 576, 625, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

Seiichi Manyama, Sep 18 2018

Intersection of A000290 and A029742. - Felix Fröhlich, Sep 18 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000290, A002113, A002779, A029742, A035090, A319389.

[n^2: n in [0..60] | not Intseq(n^2) eq Reverse(Intseq(n^2))]; // Vincenzo Librandi, Sep 19 2018

ispali:= proc(n) local L;
L:= convert(n,base,10);
L = ListTools:-Reverse(L)
end proc:
remove(ispali, [seq(i^2,i=1..100)]); # Robert Israel, Sep 18 2018

 pb10Q[n_]:=!Module[{idn10=IntegerDigits[n, 10]}, idn10==Reverse[idn10]]; Select[Range[0, 3100]^2, pb10Q] (* Vincenzo Librandi, Sep 19 2018 *)

terms(n) = my(i=0); for(k=0, oo, if(i==n, break); my(s=k^2, d=digits(s)); if(d!=Vecrev(d), print1(s, ", "); i++))
/* Print initial 50 terms as follows */
terms(50) \\ Felix Fröhlich, Sep 18 2018

A334391 Numbers whose only palindromic divisor is 1.

Original entry on oeis.org

1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 157, 163, 167, 169, 173, 179, 193, 197, 199, 211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 299, 307

Offset: 1

Bernard Schott, Apr 26 2020

Equivalent: Numbers such that the LCM of their palindromic divisors (A087999) is 1, or,
Numbers such that the number of palindromic divisors (A087990) is 1.
All terms are odd.
The 1st family consists of non-palindromic primes that form the subsequence A334321.
The 2nd family consists of {p^k, p prime, k >= 2} such that p^j for 1 <= j <= k is not a palindrome {169 = 13^2, 289 = 17^2, 361 = 19^2, ..., 2197 = 13^3, ...} (see examples).
The 3rd family consists of products p_1^q_1 * ... * p_k^q_k with k >= 2, all of whose divisors are nonpalindromic {221 = 13 * 27, 247 = 13 * 19, 299 = 13 * 23, 377 = 13 * 29, 391 = 17 * 23, 403 = 13 * 31, 481 = 13 * 37, ...}.
Also, equivalent: numbers all of whose divisors > 1 are nonpalindromic (A029742). - Bernard Schott, Jul 14 2022

			49 = 7^2, the divisor 7 is a palindrome so 49 is not a term.
169 = 13^2, divisors of 169 are {1, 13, 169} and 169 is a term.
391 = 17*23, divisors of 391 are {1,17,23,391} and 391 is a term.
307^2 = 94249 that is palindrome, so 94249 is not a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

A334321 is a subsequence.
Cf. A008365, A087990, A087999, A334139.
Cf. A029742, A087991, A093037, A355695.

notpali:= proc(n) local L;
  L:= convert(n,base,10);
  L <> ListTools:-Reverse(L)
end proc:
filter:= proc(n) option remember; andmap(notpali,numtheory:-divisors(n) minus {1}) end proc:
select(filter, [seq(i,i=1..400,2)]); # Robert Israel, Apr 28 2020

Select[Range[300], !AnyTrue[Rest @ Divisors[#], PalindromeQ] &] (* Amiram Eldar, Apr 26 2020 *)

ispal(n) = my(d=digits(n)); d == Vecrev(d);
isok(n) = fordiv(n, d, if (d>1 && ispal(d), return(0))); return(1); \\ Michel Marcus, Apr 26 2020

from sympy.ntheory import divisors, is_palindromic
def ok(n): return not any(is_palindromic(d) for d in divisors(n)[1:])
print(list(filter(ok, range(1, 308, 2)))) # Michael S. Branicky, May 08 2021

A087990(a(n)) = 1.

A087999(a(n)) = 1.

A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

Original entry on oeis.org

18, 81, 198, 576, 675, 819, 891, 918, 1131, 1304, 1311, 1818, 1998, 2262, 2622, 3393, 3933, 4031, 4154, 4514, 4636, 6364, 8181, 8749, 8991, 9478, 12441, 14269, 14344, 14421, 15167, 15602, 16237, 18018, 18449, 18977, 19998, 20651, 23843, 24882, 26677, 26892, 27225

Offset: 1

Michel Marcus, Oct 08 2020

Palindromes (A002113) are excluded from the sequence because they obviously satisfy the condition.
Sequence is infinite since it includes 18, 1818, 181818, .... See link.
There are many cases of terms that are the repeated concatenation of integers like: 1818, 8181, 181818, ... , but also 131313131313131313131313131313 and more. See A338166.
If n is in the sequence and has d digits, and gcd(n, x) = gcd(A004086(n), x) where x = (10^((k+1)*d)-1)/(10^d-1), then the concatenation of k copies of n is also in the sequence. - Robert Israel, Oct 13 2020

			For m = 18 = 2*3^2, A338038(18) = 2 + (3+2) = 7 and for m = 81 = 3^4, A338038(81) = 7, so 18 and 81 are terms.

Michel Marcus, Table of n, a(n) for n = 1..2998
Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
Muhammet Boran, Garam Choi, Steven J. Miller, Jesse Purice, and Daniel Tsai, A characterization of prime v-palindromes, arXiv:2307.00770 [math.NT], 2023.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.
Daniel Tsai, v-palindromes: an analogy to the palindromes, arXiv:2111.10211 [math.HO], 2021.

Cf. A004086 (read n backwards), A002113, A029742 (non-palindromes), A338038, A338166.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
g:= proc(n) local t;
  add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]))
end proc:
filter:= proc(n) local r;
  if n mod 10 = 0 then return false fi;
  r:= rev(n);
  r <> n and g(r)=g(n)
end proc:
select(filter, [$1..30000]); # Robert Israel, Oct 13 2020

s[1] = 0; s[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Select[Range[30000], !Divisible[#, 10] && (r = IntegerReverse[#]) != # &&  s[#] == s[r] &] (* Amiram Eldar, Oct 08 2020 *)

f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
isok(m) = my(r=fromdigits(Vecrev(digits(m)))); (m % 10) && (m != r) && (f(r) == f(m));

A280824 Numbers with an even number of digits and with an even number of distinct digits.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 1000, 1001, 1010

Offset: 1

Ilya Gutkovskiy, Jan 08 2017

Differs from A139819 (the latter contains 100, for example). - R. J. Mathar, Jan 17 2017

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit
Index entries for 10-automatic sequences.

Cf. A000035, A001637, A029742, A043537, A055642, A280823, A280825, A280826.

isA280824 := proc(n)
    if n < 10 then
        return false;
    end if;
    dgs := convert(n,base,10) ;
    if type(nops(dgs),'even') then
        type(nops(convert(dgs,set)),'even') ;
    else
        false;
    end if;
end proc: # R. J. Mathar, Jan 17 2017

Select[Range[1010], Mod[Length[IntegerDigits[#1]], 2] == 0 && Mod[Length[Union[IntegerDigits[#1]]], 2] == 0 & ]

def ok(n): s = str(n); return len(s)%2 == 0 == len(set(s))%2
print(list(filter(ok, range(1011)))) # Michael S. Branicky, Oct 12 2021

A000035(A055642(a(n))) = 0.
A000035(A043537(a(n))) = 0.
a(n) = A029742(n) for n < 82.

A350867 Non-palindromic numbers k for which d(k) = d(R(k)), where R(k) is the reversal of k and d(k) is the number of divisors of k.

Original entry on oeis.org

13, 15, 17, 24, 26, 31, 37, 39, 42, 51, 58, 62, 71, 73, 79, 85, 93, 97, 107, 113, 115, 117, 122, 123, 129, 143, 149, 155, 157, 158, 159, 165, 167, 169, 177, 178, 179, 183, 185, 187, 199, 203, 205, 221, 226, 246, 264, 265, 285, 286, 288, 294, 302, 311, 314, 319

Offset: 1

Daniel Tsai, Feb 18 2022

			264 and 462 are non-palindromic and also d(264) = 16 = d(462), and so both are members.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000005 (d), A004086 (R).

Intersection of A029742 (non-palindromes) and A062895 (d(R(k)) = d(k)).

isok(k) = my(R = fromdigits(Vecrev(digits(k)))); R != k && numdiv(R) == numdiv(k);

from sympy import divisor_count as d
def ok(k): Rk = int(str(k)[::-1]); return Rk != k and d(k) == d(Rk)
print([k for k in range(320) if ok(k)]) # Michael S. Branicky, Feb 20 2022

A048344 a(n) * a(n)_reversed is a palindrome (and a(n) is not palindromic).

Original entry on oeis.org

12, 21, 102, 112, 122, 201, 211, 221, 1002, 1011, 1012, 1021, 1022, 1101, 1102, 1112, 1121, 1201, 1202, 1211, 2001, 2011, 2012, 2021, 2101, 2102, 2111, 2201, 10002, 10011, 10012, 10021, 10022, 10102, 10111, 10112, 10121, 10202, 10211, 11001

Offset: 1

Patrick De Geest, Feb 15 1999

Does any term in this sequence have any digit greater than 2? - Harvey P. Dale, Nov 05 2011

			E.g. 10021 * 12001 = 120262021 is a palindrome.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Patrick De Geest, Palindromic Products of Non Palindromic Integers and their Reversals

Cf. A048343.

Cf. A004086, A136522, A029742.

a048344 n = a048344_list !! (n-1)
a048344_list = filter f a029742_list where
   f x = a136522 (x * a004086 x) == 1
-- Reinhard Zumkeller, Oct 09 2011

palQ[n_]:=Module[{idn=IntegerDigits[n],ridn,idn2},ridn=Reverse[idn]; idn2 = IntegerDigits[ n FromDigits[ridn]];idn!=ridn&&idn2==Reverse[idn2]]; Select[ Range[11100],palQ] (* Harvey P. Dale, Nov 05 2011 *)
Select[Range[12000],!PalindromeQ[#]&&PalindromeQ[# IntegerReverse[#]]&] (* Harvey P. Dale, Jul 10 2023 *)

A048344_list = []
for n in range(1,10**5):
    s = str(n)
    s2 = str(n)[::-1]
    if s != s2:
        s3 = str(n*int(s2))
        if s3 == s3[::-1]:
            A048344_list.append(n) # Chai Wah Wu, Sep 08 2014

Offset corrected by Reinhard Zumkeller, Oct 09 2011

A334321 Non-palindromic primes.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349

Offset: 1

Bernard Schott, Apr 23 2020

Not the same as A052085, primes whose decimal digits are grouped together: 2, 3, 5, 7, 11 are not terms of this sequence, then the next difference occurs for prime 1013 that belongs to this sequence but not to A052085.

			97 is prime and is not a palindrome, hence 97 belongs to this sequence.

Equals A000040 \ A002385.

Intersection of A029742 and A000040.

Select[Range[350], PrimeQ[#] && !PalindromeQ[#] &] (* Amiram Eldar, Apr 23 2020 *)

isok(p) = if (isprime(p), my(d=digits(p)); d != Vecrev(d)); \\ Michel Marcus, Apr 23 2020

Formula : A087999(a(n)) = 1.

A339676 Nonpalindromic numbers that are products of repunits.

Original entry on oeis.org

161051, 1490841, 1625151, 1771561, 14921841, 15043941, 16266151, 16399251, 17876661, 19487171, 137009631, 149231841, 149352841, 150574941, 151807041, 162676151, 164140251, 165483351, 178927661, 180391761, 196643271, 214358881, 1370219631, 1371330631, 1492331841

Offset: 1

Bernard Schott, Dec 12 2020

The first term is A308365(19).

G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see link, page 98). According to this conjecture, these perfect powers are terms: {11^k, k>=4}, {111^k, k>=4}, {1111^k, k>=3}, {11111^k, k>=3}, ...

			a(1) = 161051 = 11^5.
a(2) = 1490841 = 11^2 * 111^2.
a(3) = 1625151 = 11^4 * 111.
a(4) = 1771561 = 11^6.
a(5) = 14921841 = 11^2 * 111 * 1111.

David A. Corneth, Table of n, a(n) for n = 1..10000
Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].

Intersection of A308365 and A029742.

Cf. A083278, A334131.

vec[max_] := Module[{m = Floor @ Log10[9*max + 1], r, s = {1}, s1}, r = (10^Range[2, m] - 1)/9; Do[emax = Floor@Log[r[[k]], max]; s1 = r[[k]]^Range[0, emax]; s = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &], {k, 1, m - 1}]; s]; Select[vec[1.5*10^9], !PalindromeQ[#] &] (* Amiram Eldar, Dec 12 2020 *)

A154810 Nonpalindromic numbers with binary digits only.

Original entry on oeis.org

10, 100, 110, 1000, 1010, 1011, 1100, 1101, 1110, 10000, 10010, 10011, 10100, 10110, 10111, 11000, 11001, 11010, 11100, 11101, 11110, 100000, 100010, 100011, 100100, 100101, 100110, 100111, 101000, 101001, 101010, 101011, 101100, 101110, 101111

Offset: 1

Omar E. Pol, Jan 24 2009

A154809 written in base 2.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A007088, A029742, A057148, A154809.

Map[FromDigits, Select[IntegerDigits[Range[50], 2], !PalindromeQ[#] &]] (* Paolo Xausa, Jul 24 2024 *)

def A154810(n):
    def f(x): return n+(x>>(l:=x.bit_length())-(k:=l+1>>1))-(int(bin(x)[k+1:1:-1],2)>(x&(1<Chai Wah Wu, Jul 24 2024

a(n) = A007088(A154809(n)). - Michel Marcus, Jul 24 2024

Extended by Ray Chandler, Mar 14 2010

cp's OEIS Frontend

A118959 Non-palindromic numbers which are divisible by their reversal.

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A319388 Non-palindromic squares.

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A334391 Numbers whose only palindromic divisor is 1.

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A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

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A280824 Numbers with an even number of digits and with an even number of distinct digits.

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A350867 Non-palindromic numbers k for which d(k) = d(R(k)), where R(k) is the reversal of k and d(k) is the number of divisors of k.

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A048344 a(n) * a(n)_reversed is a palindrome (and a(n) is not palindromic).

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