A348429 Perfect powers m^k, m >= 1, k >= 2 such that m and m^k both are palindromes.

1, 4, 8, 9, 121, 343, 484, 1331, 10201, 12321, 14641, 40804, 44944, 1002001, 1030301, 1234321, 1367631, 4008004, 100020001, 102030201, 104060401, 121242121, 123454321, 125686521, 400080004, 404090404, 1003003001, 10000200001, 10221412201, 12102420121, 12345654321, 40000800004

Offset

1,2

Comments

Complement of A348319 relative to the positive perfect powers A001597.

This sequence is infinite since each square (10^m+1)^2 is a term for m >= 0 and A033934 is a subsequence.

Observation: terms always contain an odd number of digits.

For k = 2, subsequence of palindromes whose square root is a palindrome is A057136 (see A057135).

For k = 3, except for 2201^3 = 10662526601, all known palindromic cubes have a palindromic rootnumber (see A002780 and A002781).

For k = 4, all known integers whose fourth power is a palindrome are also palindromes (see A056810 and subsequence A186080).

For k >= 5, G. J. Simmons conjectured there are no palindromes of the form m^k for k >= 5 and m > 1 (see Simmons link p. 98); according to this conjecture, all the terms are of the form (palindrome)^k, with 2 <= k <= 4.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..1024 (all terms with <= 40 digits)

Michael S. Branicky, Python program

Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., Vol. 3, No. 2 (1970), pp. 93-98 [Annotated scanned copy].

Example

First few terms are equal to 1, 2^2, 2^3, 3^2, 11^2, 7^3, 22^2, 11^3, 101^2, 111^2, 11^4 = 121^2, 202^2, 212^2, 1001^2, 101^3, 1111^2, 111^3.

Mathematica

Block[{n = 10^6, nn, s}, s = Select[Range[2, n], PalindromeQ]; nn = Max[s]^2; {1}~Join~Union@ Reap[Table[Do[If[PalindromeQ[m^k], Sow[m^k]], {k, 2, Log[m, nn]}], {m, s}]][[-1, -1]]] (* Michael De Vlieger, Oct 18 2021 *)

Prog

(Python) # see link for faster version
def ispal(n): s = str(n); return s == s[::-1]
def aupto(limit):
    aset, m, mm = {1}, 2, 4
    while mm <= limit:
        if 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**11)) # Michael S. Branicky, Oct 18 2021
(PARI) ispal(x) = my(d=digits(x)); d == Vecrev(d); \\ A002113
isok(m) = if (m==1, return (1)); my(p); ispal(m) && ispower(m, , &p) && ispal(p); \\ Michel Marcus, Oct 19 2021
(PARI) ispal(x) = my(d=digits(x)); d == Vecrev(d); \\ A002113
lista(nn) = {my(list = List(1)); 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. A001597, A002113, A033934, A056810, A186080, A348319, A348320.

Sequence in context: A226035 A128827 A075786 * A046450 A077271 A084093

Adjacent sequences: A348426 A348427 A348428 * A348430 A348431 A348432

Keyword

nonn,base

Author

Bernard Schott, Oct 18 2021

Status

approved