A059744 -id:A059744 - 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.

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.

A059745 Palindromic squares of sporadic type.

Original entry on oeis.org

676, 69696, 94249, 698896, 5221225, 6948496, 522808225, 617323716, 942060249, 637832238736, 1086078706801, 1615108015161, 4051154511504, 5265533355625, 9420645460249, 123862676268321, 144678292876441, 165551171155561, 900075181570009, 4099923883299904, 94206450305460249

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.
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 many more terms)
William Rex Marshall, Table up to 10^40

Cf. A059744, A016106, A028818, A060088.

a(n) = A059744(n)^2. - Hugo Pfoertner, Oct 03 2023

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

A065378 Primes p such that p^2 is a palindromic square.

Original entry on oeis.org

2, 3, 11, 101, 307, 30001253, 100111001, 110111011, 111010111, 111091111, 1011099011101, 1100011100011, 1100101010011, 1101010101011, 100110101011001, 100110990111001, 101000010000101, 101011000110101, 101110000011101

Offset: 1

Patrick De Geest, Nov 03 2001

Record prime base number of a sporadic palindromic square is 13661181333262459.

			E.g. a(6) = 900075181570009 = p^2 with p = 30001253 and prime.

Hans Havermann (via Feng Yuan), Table of n, a(n) for n = 1..61
Patrick De Geest, Palindromic Squares in bases 2 to 17
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
W. R. Marshall, Palindromic Squares
F. Yuan, Palindromic Square Numbers

Cf. A065379, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

t={}; Do[p=Prime[n]; If[FromDigits[Reverse[IntegerDigits[p^2]]]==p^2,AppendTo[t,p]],{n,10^7}]; t (* Jayanta Basu, May 11 2013 *)

Corrected by Jayanta Basu, May 11 2013

A251673 Numbers that are not palindromes, but whose squares are palindromes.

Original entry on oeis.org

26, 264, 307, 836, 2285, 2636, 22865, 24846, 30693, 798644, 1042151, 1109111, 1270869, 2012748, 2294675, 3069307, 11129361, 12028229, 12866669, 30001253, 64030648, 110091011, 111091111, 306930693, 2062386218, 2481623254, 10106064399, 10109901101, 10110911101

Offset: 1

Arkadiusz Wesolowski, Dec 06 2014

The corresponding sequence excluding numbers in A059744 starts: 1109111, 110091011, 111091111, 10109901101, 10110911101, ....

The sequence is infinite, for instance it contains 111*100^k + 91*10^k + 111 for k > 3. - Emmanuel Vantieghem, Sep 30 2017

Supersequence of A059744. Cf. A029742, A002778.

[n: n in [0..3069307] | not Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(n^2, 10) eq Reverse(Intseq(n^2, 10))];

a251673[n_Integer] := Select[Range[n], IntegerDigits[#] != Reverse@IntegerDigits[#] && IntegerDigits[#^2] == Reverse@IntegerDigits[#^2] &]; a251673[10^7] (* Michael De Vlieger, Dec 14 2014 *)

for(n=1,10^6,d=digits(n);d2=digits(n^2);if(Vecrev(d2)==d2&&Vecrev(d)!=d,print1(n,", "))) \\ Derek Orr, Dec 13 2014

A029742 INTERSECT A002778.

A065379 Palindromic squares with a prime root.

Original entry on oeis.org

4, 9, 121, 10201, 94249, 900075181570009, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121, 1210222232227222322220121

Offset: 1

Patrick De Geest, Nov 03 2001

Record sporadic palindromic square with a prime root is 186627875420278656872024578726681.

			a(6) = 900075181570009 = p^2 with p = 30001253, a prime.

Patrick De Geest, Palindromic Squares in bases 2 to 17
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
William Rex Marshall, Palindromic Squares [Internet Archive Wayback Machine]

Cf. A065378, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

A258382 Non-palindromic numbers n such that the square root of n multiplied by the reversal of n is a palindrome.

Original entry on oeis.org

144, 441, 1584, 4851, 10404, 12544, 14544, 14884, 15984, 27648, 40401, 44521, 44541, 48841, 48951, 84672, 114444, 137984, 144144, 159984, 409739, 441441, 444411, 489731, 489951, 937904, 1004004, 1022121, 1024144, 1042441, 1044484, 1050804

Offset: 1

Pieter Post, May 28 2015

This sequence is infinite, because it contains several infinite subsequences such as: sqrt(1584*4851)=2772, sqrt(15984*48951)=27972, sqrt(159..984*489...951)=279...972.

It appears that the first (or last) digit is never 5, 6 or 7.

			27648 is in the sequence because sqrt(27648*84672)=48384.

Pieter Post and Giovanni Resta, Table of n, a(n) for n = 1..1124 (first 246 terms from Pieter Post, terms < 10^12)

Cf. A002113, A002778, A059744, A004086, A117281.

palQ[n_] := Block[{d = IntegerDigits@ n}, And[IntegerQ@ n, d == Reverse@ d]]; Select[Range@ 100000, And[! palQ@ #, palQ[Sqrt[# FromDigits@ Reverse@ IntegerDigits@ #]]] &] (* Michael De Vlieger, May 28 2015 *)

rev(k) = subst(Polrev(digits(k)), x, 10);
isok(n) = {rn = rev(n); if (rn != n, nrn = n*rn; issquare(nrn) && (y=sqrtint(nrn)) && (y == rev(y)););} \\ Michel Marcus, May 29 2015

for n in range (1, 10**9):
    y=int(str(n)[::-1])
    ya=int(pow(n*y,1/2))
    if ya==int(str(ya)[::-1]) and n*y==ya**2 and n!=y:
        print (n)

Numbers n such that sqrt(n*reversal(n)) is a palindrome, where n is not a palindrome.

cp's OEIS Frontend

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

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

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A059745 Palindromic squares of sporadic type.

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A060088 Palindromic squares with an asymmetric square root.

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A065378 Primes p such that p^2 is a palindromic square.

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A251673 Numbers that are not palindromes, but whose squares are palindromes.

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A065379 Palindromic squares with a prime root.

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A258382 Non-palindromic numbers n such that the square root of n multiplied by the reversal of n is a palindrome.

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