

A177952


a(n) = number of ndigit squares in base 10 such that there is at least one permutation that is also a square in base 10. Initial zeros are not allowed for any square.


0



0, 0, 7, 13, 86, 293, 1212, 4699, 17380, 60623, 203799, 664953, 2135649, 6800449, 21572602, 68311990, 216144075, 683666674
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OFFSET

1,3


COMMENTS

The ratio of a(n) to the nth entry of sequence A049415 goes to 1. Bilisoly (2008), listed below, has a proof of this. Squares of this type are called "anasquares" in this reference (short for "anagram of squares").


REFERENCES

R. Bilisoly, Square anagrams of squares, The Mathematical Gazette, 92(2008), 5863.


LINKS

Table of n, a(n) for n=1..18.


EXAMPLE

For instance, a(3) = 7 because (1) 144, 441 are both squares and permutations of each other as is 256, 625 and 169, 196, 961 and (2) there are no other 3 digit squares that can be permuted to another square (because initial zeros are forbidden, 100 and 001, etc., do not count).


MATHEMATICA

nAnasquares[ndigits_] := Module[{nsquares = 0, nkeys = 0, nanapat = 0, upper, lower, square, key, dictionary}, lower = Sqrt[10^(ndigits  1)] // Ceiling; upper = Sqrt[10^ndigits  1] // Floor; Do[ ++nsquares; square = i^2; key = ToString[FromDigits[Sort[IntegerDigits[square]]]]; If[StringQ[ dictionary[ key]] && (Length[StringPosition[dictionary[key], ", "]] == 0), ++nanapat, Null] If[StringQ[dictionary[key]], dictionary[key] = dictionary[key] <> ", " <> ToString[square], dictionary[key] = ToString[square]; ++nkeys], {i, lower, upper}]; Return[nsquares  nkeys + nanapat] ] Table[nAnasquares[n], {n, 1, 10}]


CROSSREFS

a(n) converges to A049415 in the sense that the ratio of the two sequences goes to 1 as n goes to infinity.
Sequence in context: A061521 A073220 A110455 * A132373 A110293 A253333
Adjacent sequences: A177949 A177950 A177951 * A177953 A177954 A177955


KEYWORD

base,more,nonn


AUTHOR

Roger Bilisoly (bilisolyr(AT)ccsu.edu), May 15 2010


EXTENSIONS

a(16)a(18) from Donovan Johnson, Jun 10 2010


STATUS

approved



