A029742 -id:A029742 - cp's OEIS frontend

A251673 Numbers that are not palindromes, but whose squares are palindromes.

26, 264, 307, 836, 2285, 2636, 22865, 24846, 30693, 798644, 1042151, 1109111, 1270869, 2012748, 2294675, 3069307, 11129361, 12028229, 12866669, 30001253, 64030648, 110091011, 111091111, 306930693, 2062386218, 2481623254, 10106064399, 10109901101, 10110911101

Offset: 1

Arkadiusz Wesolowski, Dec 06 2014

The corresponding sequence excluding numbers in A059744 starts: 1109111, 110091011, 111091111, 10109901101, 10110911101, ....

The sequence is infinite, for instance it contains 111*100^k + 91*10^k + 111 for k > 3. - Emmanuel Vantieghem, Sep 30 2017

Supersequence of A059744. Cf. A029742, A002778.

[n: n in [0..3069307] | not Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(n^2, 10) eq Reverse(Intseq(n^2, 10))];

a251673[n_Integer] := Select[Range[n], IntegerDigits[#] != Reverse@IntegerDigits[#] && IntegerDigits[#^2] == Reverse@IntegerDigits[#^2] &]; a251673[10^7] (* Michael De Vlieger, Dec 14 2014 *)

for(n=1,10^6,d=digits(n);d2=digits(n^2);if(Vecrev(d2)==d2&&Vecrev(d)!=d,print1(n,", "))) \\ Derek Orr, Dec 13 2014

A029742 INTERSECT A002778.

A319440 Squares of non-palindromic number.

Original entry on oeis.org

100, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3136, 3249, 3364, 3481, 3600, 3721, 3844

Offset: 1

Seiichi Manyama, Sep 19 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000290, A002113, A029742, A319388, A319441.

[n^2: n in [0..65] | Intseq(n) ne Reverse(Intseq(n))]; // Vincenzo Librandi, Sep 19 2018

palQ[n_]:=Module[{idn=IntegerDigits[n]}, idn==Reverse[idn]]; DeleteCases[Range[10, 110],?palQ]^2 (* _Vincenzo Librandi, Sep 19 2018 *)

def A319440(n):
    def f(x): return n+x//10**((l:=len(s:=str(x)))-(k:=l+1>>1))-(int(s[k-1::-1])>x%10**k)+10**(k-1+(l&1^1))-1
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**2 # Chai Wah Wu, Oct 28 2024

a(n) = A029742(n)^2.

A319441 Cubes of non-palindromic numbers.

Original entry on oeis.org

1000, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 91125, 97336, 103823, 110592, 117649, 125000, 132651, 140608

Offset: 1

Seiichi Manyama, Sep 19 2018

This is not a subsequence of A029742. - Bruno Berselli, Sep 19 2018

			2201^3 = 10662526601 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
G. J. Simmons, Palindrome cubes: Problem B-183, Fibonacci Quart. 8 (1970), no. 5, p. 551.

Cf. A000578, A002113, A029742, A319389, A319440.

[n^3: n in [0..65] | Intseq(n) ne Reverse(Intseq(n))]; // Vincenzo Librandi, Sep 19 2018

palQ[n_]:=Module[{idn=IntegerDigits[n]}, idn==Reverse[idn]]; DeleteCases[Range[10, 110], ?palQ]^3 (* _Vincenzo Librandi, Sep 19 2018 *)
Select[Range[100],!PalindromeQ[#]&]^3 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2019 *)

is_a029742(n)=my(d=digits(n)); d!=Vecrev(d) \\ after Charles R Greathouse IV in A029742
terms(n) = my(i=0, x=1); while(1, if(i==n, break, if(is_a029742(x), print1(x^3, ", "); i++)); x++)
/* Print initial 40 terms as follows */
terms(40) \\ Felix Fröhlich, Sep 19 2018

def A319441(n):
    def f(x): return n+x//10**((l:=len(s:=str(x)))-(k:=l+1>>1))-(int(s[k-1::-1])>x%10**k)+10**(k-1+(l&1^1))-1
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m**3 # Chai Wah Wu, Jul 24 2024

a(n) = A029742(n)^3.

A357044 Lexicographic earliest sequence of distinct palindromes (A002113) such that a(n)+a(n+1) is never palindromic.

Original entry on oeis.org

1, 9, 3, 7, 5, 8, 2, 11, 4, 6, 22, 88, 44, 66, 77, 33, 99, 55, 101, 909, 111, 191, 121, 181, 131, 171, 141, 161, 151, 252, 262, 242, 272, 232, 282, 222, 292, 212, 393, 313, 494, 323, 383, 333, 373, 343, 363, 353, 454, 464, 444, 474, 434, 484, 424

Offset: 1

Eric Angelini and M. F. Hasler, Sep 14 2022

Obviously the sequence cannot contain 0.

It is easy to prove that the sequence is a permutation of the nonzero palindromes (in the sense that it contains each of them exactly once).

Eric Angelini, Sums with palindromes, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022
Eric Angelini, Sums with palindromes, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022 [Cached copy]

Cf. A002113 (palindromes), A029742 (non-palindromes), A262038 (next palindrome), A357045 (non-palindromes with palindromic sum of neighbors).

A357044_first(n, U=[0], a=9)={vector(n,k, k=U[1]; while(is_A002113(a+k=A262038(k+1)) || setsearch(U, k), ); U=setunion(U,[a=k]); while(#U>1 && U[2]==A262038(U[1]+1), U=U[^1]); a)}

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def nextpal(p): # next largest palindrome after palindrome p
    d = str(p)
    if set(d) == {'9'}: return int('1' + '0'*(len(d)-1) + '1')
    h = str(int(d[:(len(d)+1)//2]) + 1)
    return int(h + h[:-1][::-1]) if len(d)&1 else int(h + h[::-1])
def agen():
    aset, pal, minpal = {1}, 1, 2
    while True:
        an = pal; yield an; aset.add(an); pal = minpal
        while pal in aset or ispal(an+pal): pal = nextpal(pal)
        while minpal in aset: minpal = nextpal(minpal)
print(list(islice(agen(), 55))) # Michael S. Branicky, Sep 14 2022

A062908 Non-palindromic number and its reversal are both even.

Original entry on oeis.org

20, 24, 26, 28, 40, 42, 46, 48, 60, 62, 64, 68, 80, 82, 84, 86, 200, 204, 206, 208, 210, 214, 216, 218, 220, 224, 226, 228, 230, 234, 236, 238, 240, 244, 246, 248, 250, 254, 256, 258, 260, 264, 266, 268, 270, 274, 276, 278, 280, 284, 286, 288, 290, 294, 296

Offset: 1

Amarnath Murthy, Jul 01 2001

			24 and 42 are both multiples of 2.

Cf. A029742 (non-palindromic), A005843 (even numbers).

n := 2; stop := 410; m := 0; while m < stop do rev := int_reverse(m); if m <> rev and rev mod n = 0 then write(m," "); end; inc(m,n); end;

Select[Range[2,296,2],EvenQ[Last[x=Reverse[y=IntegerDigits[#]]]] && x!=y &] (* Jayanta Basu, May 17 2013 *)

isok(m) =  {if (!(m%2), my(r=fromdigits(Vecrev(digits(m)))); if ((r!=m) && !(r%2), print1(m, ", ")););} \\ Michel Marcus, Oct 10 2020

More terms from Dean Hickerson, Jul 06 2001

A217252 Lucky numbers whose digital reversal is another lucky number.

Original entry on oeis.org

13, 15, 31, 37, 51, 73, 115, 133, 163, 169, 189, 193, 195, 327, 331, 339, 361, 385, 391, 399, 511, 529, 537, 579, 583, 591, 723, 729, 735, 739, 925, 927, 933, 937, 961, 975, 981, 993

Offset: 1

M. F. Hasler, Mar 16 2013

Inspired by the error in A140291 (forgotten palindromes 33 and 99), pointed out by L. Edson Jeffery in a post to the SeqFan list.
This sequence is obtained from A118561 by removal of the palindromes A031161 = (1, 3, 7, 9, 33, 99, 111, 141, 151, 171, 303, 393, 535, 717, 727, 777, 787, 979, ...)
By analogy with the "emirps" A006567 and "emirpimes" A097393, these might be called "ykcul"s, for lucky numbers which, when reversed, are different lucky numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000959, A002385, A006567, A097393, A029742, A031161, A002113, A118561, A140291.

A217252 = A118561 \ A031161 = A118561 \ A002113 = A118561 intersect A029742 = { n in A000959 | R(n) is in A000959 and different from n }, where R = A004086.

A280354 Numbers n such that (i) number of divisors of n equals number of divisors of digit reversal of n, (ii) sum of divisors of n equals sum of divisors of digit reversal of n, and (iii) n is not a palindrome.

Original entry on oeis.org

1561, 1651, 5346, 6435, 157661, 166751, 301134, 321853, 358123, 431103, 507955, 511665, 517055, 537495, 539946, 550715, 559705, 566115, 576908, 594735, 649935, 729287, 765677, 776567, 782927, 809675, 834498, 894438, 896898, 898698, 905289, 982509, 1257912, 1473302

Offset: 1

Ilya Gutkovskiy, Jan 01 2017

Intersection of A062895 and A085329.

Numbers n such that A000005(n) = A000005(A004086(n)), A000203(n) = A000203(A004086(n)) and A136522(n) = 0.

			1561 is in the sequence because 1561 has 4 divisors {1, 7, 223, 1561}, 1 + 7 + 223 + 1561 = 1792 and 1651 has 4 divisors {1, 13, 127, 1651}, 1 + 13 + 127 + 1651 = 1792.

Indranil Ghosh, Table of n, a(n) for n = 1..200
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A004086, A029742, A062895, A085329, A136522.

Select[Range[1500000], !PalindromeQ[#1] && DivisorSigma[0, #1] == DivisorSigma[0, FromDigits[Reverse[IntegerDigits[#1]]]] && DivisorSigma[1, #1] == DivisorSigma[1,FromDigits[Reverse[IntegerDigits[#1]]]] & ]
fQ[n_]:=With[{irn=IntegerReverse[n]},!PalindromeQ[n]&&DivisorSigma[0,n]==DivisorSigma[0,irn] && DivisorSigma[1,n] == DivisorSigma[ 1,irn]]; Select[Range[1480000],fQ] (* Harvey P. Dale, Dec 17 2024 *)

R(n) = eval(concat(Vecrev(Str(n))));
isok(n) = n != R(n) && numdiv(n) == numdiv(R(n)) && sigma(n) == sigma(R(n));
for(n=1561, 1473302, if(isok(n), print1(n, ", "))) \\ Indranil Ghosh, Mar 06 2017

A359510 Numbers that can't be written as a palindromic product, i.e., such that the concatenation of all digits of the factors yields a palindrome.

Original entry on oeis.org

23, 26, 29, 30, 34, 35, 37, 38, 43, 47, 53, 57, 59, 62, 65, 67, 70, 73, 74, 79, 82, 83, 85, 86, 87, 89, 92, 94, 95, 97, 103, 106, 107, 109, 123, 127, 130, 134, 137, 139, 140, 142, 145, 146, 148, 149, 152, 157, 158, 163, 167, 170, 173, 174, 178, 179, 182, 183, 185, 190, 193, 194, 197

Offset: 1

M. F. Hasler and Eric Angelini, Jan 03 2023

Any number of factors 1 is allowed anywhere in the product.

The sequence contains all primes which are not palindromic when stripped of digits '1' on either side (for example 23, 29, 37, but not 13, 17, 19, 31 which can be written as 13*1, 17*1, 19*1, 1*31, etc., where the concatenation of all digits, "131", "171", ... is palindromic).

			Any palindrome is trivially a palindromic product and therefore not in the sequence. Also not in the sequence are 10 = 10*1, 12 = 12*1, ..., 20 = 2*5*2, 21 = 1*21. Therefore the first term is a(1) = 23.

Eric Angelini, 2023 = 7*17*17, a palindromic product, math-fun list (restricted to subscribers), Jan. 1, 2023.

Cf. A002113 (palindromes in base 10), A029742 (non-palindromes), A334321 (non-palindromic primes), A004176 (omit digits 1).

A372488 The smallest nonpalindromic number that shares n or more distinct prime factors with the prime factors of its reverse.

Original entry on oeis.org

10, 12, 24, 264, 8580, 24024, 2168166, 67897830, 2448684420

Offset: 0

Scott R. Shannon, May 02 2024

When a number is reversed any leading 0's are dropped; the resulting number may be palindromic.

			a(3) = 264 as 264 = 2^3 * 3 * 11 and 264 in reverse is 462 = 2 * 3 * 7 * 11, which share three prime factors 2, 3, and 11.

Cf. A335645, A046399, A027746, A029742, A372384.

A377191 Smallest number of digits that must be changed in n to obtain a palindrome (without changing the first digit to 0).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 0

Franz Vrabec, Oct 19 2024

The sequence is unbounded, with a(n) = k first occurring at n = A377192(k).

			a(12) = 1 because 12 is not a palindrome, but changing 1 digit appropriately (either the first or second in this case) yields a palindrome.

Cf. A002113, A029742, A136522, A377192.

a(A002113(n)) = 0, a(A029742(n)) > 0.

a(n) = 0 iff A136522(n) = 1, a(n) > 0 iff A136522(n) = 0.

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A251673 Numbers that are not palindromes, but whose squares are palindromes.

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A319440 Squares of non-palindromic number.

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A319441 Cubes of non-palindromic numbers.

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A357044 Lexicographic earliest sequence of distinct palindromes (A002113) such that a(n)+a(n+1) is never palindromic.

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A062908 Non-palindromic number and its reversal are both even.

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A217252 Lucky numbers whose digital reversal is another lucky number.

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A280354 Numbers n such that (i) number of divisors of n equals number of divisors of digit reversal of n, (ii) sum of divisors of n equals sum of divisors of digit reversal of n, and (iii) n is not a palindrome.

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A359510 Numbers that can't be written as a palindromic product, i.e., such that the concatenation of all digits of the factors yields a palindrome.

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