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A348319 Perfect powers m^k, k >= 2 that are palindromes while m is not a palindrome. 2
676, 69696, 94249, 698896, 5221225, 6948496, 522808225, 617323716, 942060249, 10662526601, 637832238736, 1086078706801, 1230127210321, 1615108015161, 4051154511504, 5265533355625, 9420645460249, 123862676268321, 144678292876441, 165551171155561, 900075181570009 (list; graph; refs; listen; history; text; internal format)
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
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. A059745 (a subsequence).
Subsequence of A001597 and of A075786.
Sequence in context: A239608 A238252 A264338 * A059745 A028818 A203541
KEYWORD
nonn,base
AUTHOR
Bernard Schott, Oct 12 2021
EXTENSIONS
a(18)-a(21) from Amiram Eldar, Oct 12 2021
STATUS
approved

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Last modified June 24 15:53 EDT 2024. Contains 373679 sequences. (Running on oeis4.)