A060087 -id:A060087 - cp's OEIS frontend

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

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

A007573 a(n) is the number of base numbers with 2n+1 digits in the asymmetric families of palindromic squares.

Original entry on oeis.org

1, 2, 5, 6, 9, 10, 10, 15, 15, 16, 18, 24, 18, 26, 24, 30, 27, 33, 28, 40, 33, 40, 35, 48, 37, 50, 42, 53, 45, 58, 46, 64, 50, 64, 54, 72, 55, 73, 60, 78, 63, 82, 63, 88, 69, 88, 72, 95, 73, 98, 78, 102, 80, 106, 82, 112, 87, 111, 90, 120, 91, 122, 95, 126, 99, 130, 100, 135

Offset: 3

Simon Plouffe, N. J. A. Sloane, Robert G. Wilson v

			a(3) = 1: The only base number of length 2*3 + 1 = 7 is 1109111 = A060087(1);
a(4) = 2 indicates the existence of two length 2*4 + 1 = 9 base numbers, 110091011 = A060087(2) and 111091111 = A060087(3).

M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 3..115
Patrick De Geest, Subsets of Palindromic Squares
IBM Research Ponder This, Non-palindromic numbers with palindromic squares, October 2023 - Challenge.
M. Keith, Classification and enumeration of palindromic squares, J. Rec. Math., 22 (No. 2, 1990), 124-132. [Annotated scanned copy]. See foot of page 130.
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A002779, A060087, A060088.

PARI
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\\ see P. De Geest link.
```

a(17)-a(31) from Sean A. Irvine, Jan 10 2018
Name and offset corrected by Hugo Pfoertner, Oct 04 2023
a(32)-a(70) from Hugo Pfoertner, Oct 07 2023

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

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.

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

A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

Original entry on oeis.org

1010, 1101, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1212, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1313, 1321, 1341, 1351, 1361, 1371, 1381, 1391, 1401, 1411, 1414, 1421, 1431, 1451, 1461, 1471, 1481, 1491, 1501, 1511, 1515, 1521, 1531

Offset: 1

M. F. Hasler, May 03 2017

The pseudo- or almost-palindromic numbers considered here are not related to the similarly named but different concepts mentioned in comments on A003555 and in A060087 - A060088.
We could consider "more general" palindromic concatenations like A.B.B.A, A.B.C.B.A, etc., but all of these can be written as A.B'.A with B' = B.B resp. B.C.B, etc. The result is non-palindromic (i.e., not in A002113) as required, if and only if at least one of the strings is non-palindromic.
Here, A is allowed to have only one digit, so most of the first 100 terms are of the form 1.B.1 where B = 10, 12, 13, ... (palindromes 11, 22, 33, ... excluded).
If all of the strings A, B (...) are required to be non-palindromic, the sequence starts with terms of the form A.A with A = 10, 12, 13, ..., 98:  1010, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2323, .... This is a subsequence of A239019 (numbers which are not primitive words over the alphabet {0,...,9} when written in base 10).

A286138 = select(t->!is_A002113(t),setunion(vector(801,i,((i-1)\89+1)*1001+((i-1)%89+1)*10),vector(89,i,(i+9)*101))) \\ The first 810 terms.

cp's OEIS Frontend

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

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A007573 a(n) is the number of base numbers with 2n+1 digits in the asymmetric families of palindromic squares.

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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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A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

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Author

Keywords

Comments

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PARI