A059745 -id:A059745 - cp's OEIS frontend

A118651 Numbers k such that k^2 is a palindrome when written in base 17.

0, 1, 2, 3, 4, 6, 12, 18, 28, 36, 84, 108, 290, 307, 324, 341, 580, 597, 614, 1080, 1614, 1740, 1842, 2616, 3378, 3480, 3720, 4344, 4824, 4914, 5220, 5526, 6408, 9828, 10134, 10440, 14472, 17944, 19336, 24360, 27624, 29484, 31320, 33144, 33960

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr), May 12 2006

			E.g. 4^2 = 16_10 = G_16, 6^2 = 36_10 = 22_17, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A029984 for base 3, A029986 for base 4, A029988 for base 5, A029990 for base 6, A029992 for base 7, A029805 for base 8, A029994 for base 9, A002778 for base 10, A029996 for base 11, A029733 for base 16

Cf. also A016106, A028818, A059744, A059745.

A059744 Numbers k such that k^2 is a palindromic square of sporadic type.

Original entry on oeis.org

26, 264, 307, 836, 2285, 2636, 22865, 24846, 30693, 798644, 1042151, 1270869, 2012748, 2294675, 3069307, 11129361, 12028229, 12866669, 30001253, 64030648, 306930693, 2062386218, 2481623254, 10106064399, 10207355549, 13579355059, 22865150135, 30101273647, 30693069307

Offset: 1

N. J. A. Sloane, Feb 21 2001

C. Ashbacher, More on palindromic squares, J. Rec. Math. 22, no. 2 (1990), 133-135. [A scan of the first page of this article is included with the last page of the Keith (1990) scan]
J. K. R. Barnett, "Tables of Square Palindromes in Bases 2 and 10," Journal of Recreational Mathematics, 23:1, pp. 13-18, 1991.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 26, pp 10, Ellipses, Paris 2008.
M. Keith, "Classification and Enumeration of Palindromic Squares," Journal of Recreational Mathematics, 22:2, pp. 124-132, 1990.
R. Ondrejka, "A Palindrome (151) of Palindromic Squares," Journal of Recreational Mathematics, 20:1, pp. 68-71, 1988.

Patrick De Geest, Full List of the Sporadic Square Palindromes (SSP), part of Palindromic Squares website on WorldOfNumbers.
M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy] (Has ten more terms)
W. R. Marshall, Table up to 10^20

Cf. A059745, A016106, A028818, A060087.

Select[Range[1042151], ! PalindromeQ[#] && PalindromeQ[#^2] &] (* Michael De Vlieger, Oct 03 2023, not suitable for terms > 1042151, needs amendment for larger terms *)

More terms from WorldOfNumbers website, communicated by Hugo Pfoertner, Oct 03 2023

A060087 Numbers k such that k^2 is a palindromic square with an asymmetric root.

Original entry on oeis.org

1109111, 110091011, 111091111, 10109901101, 10110911101, 11000910011, 11010911011, 11100910111, 1010099010101, 1010109110101, 1011099011101, 1100009100011, 1101009101011, 1110009100111, 100109990011001

Offset: 1

Patrick De Geest, Feb 15 2001

With 'asymmetric' is meant almost palindromic with a 'core' (pseudo-palindromic). The core '09' when transformed into '1n' (n=-1) makes the base number palindromic. E.g., 1109111 is in fact 11_09_111 -> 11_(10-1)_111 -> 11_1n_111 -> 111n111 and palindromic. Similarly core 099 becomes 10n, core 0999 becomes 100n, etc.

M. Keith, "Classification and Enumeration of Palindromic Squares," Journal of Recreational Mathematics, 22:2, pp. 124-132, 1990.

Hugo Pfoertner, Table of n, a(n) for n = 1..10032
Patrick De Geest, Subsets of Palindromic Squares
IBM Research Ponder This, Non-palindromic numbers with palindromic squares, October 2023 - Challenge.

Cf. A060088, A007573, A059744, A059745.

A060088 Palindromic squares with an asymmetric square root.

Original entry on oeis.org

1230127210321, 12120030703002121, 12341234943214321, 102210100272001012201, 102230523292325032201, 121020021070120020121, 121240161292161042121, 123230205292502032321, 1020300010207020100030201

Offset: 1

Patrick De Geest, Feb 15 2001

M. Keith, "Classification and Enumeration of Palindromic Squares," Journal of Recreational Mathematics, 22:2, pp. 124-132, 1990.

P. De Geest, Subsets of Palindromic Squares

Cf. A060087, A007573, A059744, A059745.

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

cp's OEIS Frontend

A118651 Numbers k such that k^2 is a palindrome when written in base 17.

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A059744 Numbers k such that k^2 is a palindromic square of sporadic type.

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A060087 Numbers k such that k^2 is a palindromic square with an asymmetric root.

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A060088 Palindromic squares with an asymmetric square root.

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

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