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

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

Offset

1,1

Comments

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.

References

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

Links

Table of n, a(n) for n=1..21.

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]

Example

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

Mathematica

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 *)

Prog

(Python)
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
(PARI) 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

Crossrefs

Cf. A002113, A002780, A028818, A060087, A251673, A348320.

Cf. A059745 (a subsequence).

Subsequence of A001597 and of A075786.

Sequence in context: A239608 A238252 A264338 * A059745 A028818 A203541

Adjacent sequences: A348316 A348317 A348318 * A348320 A348321 A348322

Keyword

nonn,base

Author

Bernard Schott, Oct 12 2021

Extensions

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

Status

approved