A075786 -id:A075786 - cp's OEIS frontend

A076443 Even-digit palindromic perfect powers.

1331, 698896, 1003003001, 637832238736, 1000030000300001, 1033394994933301, 1331399339931331, 4099923883299904, 1000000300000030000001, 1003303931991393033001, 1030331909339091330301, 1331003993003993001331, 6916103777337773016196

Offset: 1

Robert G. Wilson v, Oct 12 2002

			a(1) = 11^3; a(2) = (4*11*19)^2; a(3) = (7*11*13)^3; a(4) = (4*7*11*2593)^2; a(5) = (11*9091)^3, a(6) = (7*11*13*101)^3, a(7) = (11*73*137)^3, etc.

Chai Wah Wu, Table of n, a(n) for n = 1..40

Cf. A001597, A075786.

a = {}; Do[ q = IntegerDigits[n]; p = FromDigits[ Join[q, Reverse[q]]]; If[ Apply[ GCD, Last[ Transpose[ FactorInteger[p]]]] > 1, a = Append[a, p]; Print[p]], {n, 1, 4000000}]

a(8)-a(13) from Donovan Johnson, Oct 03 2011

A342942 Numbers whose palindromization is a perfect power.

Original entry on oeis.org

12, 13, 34, 48, 67, 102, 123, 146, 408, 449, 696, 698, 942, 1002, 1030, 1234, 1367, 4008, 5221, 6948, 10002, 10030, 10203, 10406, 12124, 12345, 12568, 40008, 40409, 52280, 61732, 94206, 100002, 102214, 106625, 121024, 123456, 400008, 637832, 1000002, 1000300, 1002003

Offset: 1

Michel Marcus, Mar 30 2021

Palindromization is the function that extends the string representation of a number into a palindrome.

Even palindromization is the concatenation of a number and its reversal. Odd palindromization excludes the first digit of the reversal.

			12 is a term because 121 is a square.
13 is a term because 1331 is a cube.

Chai Wah Wu, Table of n, a(n) for n = 1..1997 (terms 1..100 from Michel Marcus)

Cf. A075786, A076443, A001597, A002113, A342803.

Select[Range[10,50000],Or@@(GCD@@Last/@FactorInteger@#>1&/@FromDigits/@(Join[a,Reverse@#]&/@{a=IntegerDigits@#,Most@a}))&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

rev(x) = strjoin(Vecrev(Str(x)));
isok(m) = ispower(eval(Str(m, rev(m)))) || ispower(eval(Str(m, rev(m\10))));

A128827 Perfect powers beginning and ending with the same digit.

Original entry on oeis.org

1, 4, 8, 9, 121, 343, 484, 676, 1331, 1521, 1681, 4624, 5625, 9409, 10201, 11881, 12321, 14161, 14641, 16641, 17161, 19321, 19881, 21952, 40804, 43264, 44944, 47524, 49284, 50625, 55225, 60516, 64516, 65536, 69696, 79507, 91809, 94249

Offset: 1

J. M. Bergot, Apr 12 2007

			97^2 = 9409 is a term; 43^3 = 79507 is a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001597 (perfect powers), A075786 (palindromic perfect powers), A129450.

PP:=[1] cat [ n: n in [2..100000] | IsPower(n) ]; [ n: n in PP | k[1] eq k[ #k] where k is Intseq(n, 10) ]; // Klaus Brockhaus, Apr 16 2007

Join[{1},Select[Range[2,10^5],First[IntegerDigits[#]]==Last[IntegerDigits[#]]&&ResourceFunction["PerfectPowerQ"][#]&]] (* James C. McMahon, Jan 11 2025 *)

Edited, corrected and extended by Klaus Brockhaus, Apr 16 2007

A342803 Primes p whose palindromization A082216(p) is a square or higher power.

Original entry on oeis.org

67, 449, 1367, 10303, 12343, 1003003, 1022141, 1230127, 1234543, 4004009, 121200307, 10022234347, 10201204021, 10203242527, 12100242001, 13310399303, 16151080151, 52281509069, 61584539747, 90608667517, 104190107303, 1020102040201, 1022143262341, 12384043938083

Offset: 1

Lamine Ngom, Mar 22 2021

Palindromization is the function that minimally extends the string representation of a number into a palindrome (see A082216).
Are 13 and 1367 the unique terms leading to cubes or higher powers?
It seems that 13 is the unique prime whose even palindromization (the concatenation of a number and its reversal) is a square or higher power.
The next term (if it exists) is greater than 10^17.

			The prime 449 belongs to sequence because 44944 is a square: 212^2.
The prime 1367 is in the sequence since 1367631 is a cube: 111^3.
The prime 13 is not a term as A082216(13) = 131 and 131 is prime. The prime 10303 is in the sequence since 1030301 is a cube: 101^3. - _Chai Wah Wu_, Aug 26 2021

Chai Wah Wu, Table of n, a(n) for n = 1..215

Cf. A075786, A076443, A001597, A002113.
Cf. A082216 (smallest palindrome beginning with n).
Subsequence of primes of A342942.

Select[Prime@Range@100000,Or@@(GCD@@Last/@FactorInteger@#>1&/@(FromDigits/@(Join[a,Reverse@#]&/@{a=IntegerDigits@#,Most@a})))&] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

Corrected terms and missing terms added by Chai Wah Wu, Aug 26 2021

A348319 Perfect powers m^k, k >= 2 that are palindromes while m is not a palindrome.

Original entry on oeis.org

676, 69696, 94249, 698896, 5221225, 6948496, 522808225, 617323716, 942060249, 10662526601, 637832238736, 1086078706801, 1230127210321, 1615108015161, 4051154511504, 5265533355625, 9420645460249, 123862676268321, 144678292876441, 165551171155561, 900075181570009

Offset: 1

Bernard Schott, Oct 12 2021

Seems to be the "converse" of A348320.
The first nine terms are the first nine palindromic squares of sporadic type (A059745). Then, a(10) = 10662526601 = 2201^3 is the only known palindromic cube whose root is not palindromic (see comments in A002780 and Penguin reference).
The first square that is not in A059745 is a(13) = 1230127210321 = 1109111^2 = A060087(1)^2 since it is a palindromic square that is not of sporadic type, but with an asymmetric root. Indeed, all the squares of terms in A060087 are terms of this sequence (see Keith link).
Also, all the squares of terms in A251673 are terms of this sequence.
G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons link p. 98), according to this conjecture, we have 2 <= k <= 4.

			676 = 26^2, 10662526601 = 2201^3, 12120030703002121 = 110091011^2 are terms.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 10662526601, page 188.

Michael Keith, Classification and enumeration of palindromic squares, J. Rec. Math., Vol. 22, No. 2 (1990), pp. 124-132. [Annotated scanned copy]. See foot of page 130.
Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., Vol. 3, No. 2 (1970), pp. 93-98 [Annotated scanned copy]

Cf. A002113, A002780, A028818, A060087, A251673, A348320.
Cf. A059745 (a subsequence).
Subsequence of A001597 and of A075786.

seq[max_] := Module[{m = Floor@Sqrt[max], s = {}, n, p}, Do[If[PalindromeQ[k], Continue[]]; n = Floor@Log[k, max]; Do[If[PalindromeQ[(p = k^j)], AppendTo[s, p]], {j, 2, n}], {k, 1, m}]; Union[s]]; seq[10^10] (* Amiram Eldar, Oct 12 2021 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d); \\ A002113
lista(nn) = {my(list = List()); for (k=2, sqrtint(nn), if (!ispal(k), my(q = k^2); until (q > nn, if (ispal(q), listput(list, q)); q *= k;););); vecsort(list,,8);} \\ Michel Marcus, Oct 20 2021

def ispal(n): s = str(n); return s == s[::-1]
def aupto(limit):
    aset, m, mm = set(), 10, 100
    while mm <= limit:
        if not ispal(m):
            mk = mm
            while mk <= limit:
                if ispal(mk): aset.add(mk)
                mk *= m
        mm += 2*m + 1
        m += 1
    return sorted(aset)
print(aupto(10**13)) # Michael S. Branicky, Oct 12 2021

a(18)-a(21) from Amiram Eldar, Oct 12 2021

A129450 a(n) = smallest perfect power that begins and ends with digit n, 1 <= n <= 9; one-digit numbers are excluded.

Original entry on oeis.org

121, 21952, 343, 484, 5625, 676, 79507, 8242408, 9409

Offset: 1

J. M. Bergot, Apr 12 2007

			a(8) = 202^3 = 8242408; there is no smaller perfect power that begins and ends with digit 8.

Cf. A001597 (perfect powers), A075786 (palindromic perfect powers), A128827.

PP:=[1] cat [ n: n in [2..9000000] | IsPower(n) ]; firstlast:=function(x); for n:=1 to #PP do k:=Intseq(PP[n], 10); if #k gt 1 and k[1] eq x and k[ #k] eq x then return PP[n]; end if; end for; return -1; end function; [ firstlast(d): d in [1..9] ]; // Klaus Brockhaus, Apr 16 2007

Edited, corrected and extended by Klaus Brockhaus, Apr 16 2007

A384612 a(n) is the smallest integer k such that k^n is an abelian square; or -1 if no such k exists.

Original entry on oeis.org

11, 836, 11, 207, 624, 818222, 1001, 2776, 100001, 32323107, 100001, 85692627, 10000001, 501249084, 10000001, 27962757, 41695607, 70983559, 72768046, 977688137, 219873071, 112562383, 2338280974, 2435385853, 1231380445, 4557057314, 361499019, 8096434047, 5278552513

Offset: 1

Gonzalo Martínez, Jun 04 2025

Terms are the base of the smallest n-th power whose string of decimal digits is an abelian square; i.e., of the form m concatenated with a permutation of m (A272655).
If n is odd and A001700((n-1)/2) has d digits, then 0 < k <= 10^(2*d-1) + 1. - Robert Israel, Jun 05 2025
a(23) >= 1.83 * 10^9. Using Robert Israel's comment above a(23) <= 10^13 + 1. - David A. Corneth, Jun 06 2025

			a(1) = 11, since 11^1  = 1|1
a(2) = 836, since 836^2 = 698|896
a(3) = 11, since 11^3 = 13|31
a(4) = 207, since 207^4 = 18360|36801
a(5) = 624, since 624^5 = 9460692|9690624
a(6) = 818222, since 818222^6 = 300072996174564185|100579862765194304
a(7) = 1001, since 1001^7 = 10070210350|35021007001
a(8) = 2776, since 2776^8 = 35265958674713|24535718936576
a(9) = 100001, since 100001^9 = 10000900036000840012600|12600084000360000900001.

Chai Wah Wu, Table of n, a(n) for n = 1..34

Cf. A001597, A001700, A075786, A272655, A342942.

f:= proc(n) local k,d,x,L;
      k:= 2;
      do
        x:= k^n;
        d:= ilog10(x)+1;
        if d::odd then k:= ceil(10^(d/n)); next fi;
        L:= convert(x,base,10);
        if sort(L[1..d/2]) = sort(L[d/2+1..d]) then return k fi;
        k:= k+1
      od;
end proc:
map(f, [$1..30]); # Robert Israel, Jun 05 2025

from itertools import count
def ok(k, n):
    s = str(k**n)
    if len(s) % 2 != 0:
        return False
    mid = len(s) // 2
    return sorted(s[:mid]) == sorted(s[mid:])
def a(n):
    return next(k for k in count(2) if ok(k, n))
print([a(n) for n in range(1, 10)])

a(20)-a(22) from David A. Corneth, Jun 06 2025
a(23) from Gonzalo Martínez, Jun 06 2025
a(24)-a(29) from Jinyuan Wang, Jun 14 2025

cp's OEIS Frontend

A076443 Even-digit palindromic perfect powers.

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A342942 Numbers whose palindromization is a perfect power.

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A128827 Perfect powers beginning and ending with the same digit.

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Magma

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A342803 Primes p whose palindromization A082216(p) is a square or higher power.

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A348319 Perfect powers m^k, k >= 2 that are palindromes while m is not a palindrome.

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A129450 a(n) = smallest perfect power that begins and ends with digit n, 1 <= n <= 9; one-digit numbers are excluded.

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Magma

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A384612 a(n) is the smallest integer k such that k^n is an abelian square; or -1 if no such k exists.

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