A192993 -id:A192993 - cp's OEIS frontend

A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 164, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196

Offset: 1

Klaus Brockhaus, Jun 19 2011

Complement of A193096; A193095(a(n)) > 0; A038670, A039686, A167535, A192993, A193097 and A193144 are subsequences. [Reinhard Zumkeller, Jul 17 2011]

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A000290, A192993.

import Data.List (findIndices)
a191933 n = a191933_list !! (n-1)
a191933_list = findIndices (> 0) $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

CheckSplits:=function(n); v:=false; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then v:=true; end if; end for; return v; end function; [ p: p in [1..1200] | CheckSplits(p) ];

Take[Union[Flatten[Table[FromDigits[Flatten[{IntegerDigits[m^2], IntegerDigits[n^2]}]], {m, 20}, {n, 20}]]], 50] (* Alonso del Arte, Aug 11 2011 *)
squareQ[n_] := IntegerQ[Sqrt[n]]; okQ[n_] := MatchQ[IntegerDigits[n], {a__ /; squareQ[FromDigits[{a}]], b__ /; First[{b}] > 0 && squareQ[FromDigits[ {b}]]}]; Select[Range[2000], okQ] (* Jean-François Alcover, Dec 13 2016 *)

A193095 Number of times n can be written as concatenation of exactly two nonzero squares in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Jul 17 2011

a(A193096(n))=0; a(A191933(n))>0; a(A193097(n))=1; a(A192993(n))>1; a(A038670(n))=2.

			a(164) = 2, A191933(15) = A192993(1) = 164: 1'64 == 16'4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A010052.

a193095 n = sum $ map c [1..(length $ show n) - 1] where
   c k | head ys == '0' = 0
       | otherwise      = a010052 (read xs) * a010052 (read ys) where
       (xs,ys) = splitAt k $ show n

A193095(n) = sum( t=1,#Str(n)-1, apply(issquare,divrem(n,10^t))==[1,1]~ && n%10^t>=10^(t-1))  \\ M. F. Hasler, Jul 24 2011

A193095(n)={ my(c,p=1); while( n>p*=10, n%p*10>=p||next; issquare(n%p)||next; issquare(n\p) && c++);c}  \\ M. F. Hasler, Jul 24 2011

A038670 Concatenations of two squares in two ways.

Original entry on oeis.org

164, 1441, 1625, 1961, 2564, 4841, 12116, 14449, 16400, 25625, 46241, 48464, 115625, 116641, 144100, 148841, 160025, 162500, 163844, 169169, 184964, 193636, 196100, 256400, 361225, 368649, 466564, 484100, 493025, 961009, 973441

Offset: 1

David W. Wilson

Equals A192993 for terms < 40000000, see comment and b-file in A192993. - Reinhard Zumkeller, Jul 15 2011

Subsequence of A192993; A193095(a(n)) = 2. - Reinhard Zumkeller, Jul 17 2011

			a(1) = 164 = concat(1^2,8^2) = concat(4^2, 2^2).

import Data.List (elemIndices)
a038670 n = a038670_list !! (n-1)
a038670_list = elemIndices 2 $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

A193097 Numbers that are the concatenation of exactly one pair of nonzero squares.

Original entry on oeis.org

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196, 1211

Offset: 1

Reinhard Zumkeller, Jul 17 2011

Subsequence of A191933; A193095(a(n)) = 1.

			161 = concat(4^2,1^2), therefore 161 is a term;
164 = concat(1^2,8^2) = concat(4^2,2^2), therefore 164 is not a term (A191933(15)=A192993(1)=164, A193095(164)=2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndices)
a193097 n = a193097_list !! (n-1)
a193097_list = elemIndices 1 $ map a193095 [0..]

Take[Union[FromDigits[Flatten[IntegerDigits/@((#)^2)]]&/@Tuples[Range[14],2]],60] (* Harvey P. Dale, Jul 27 2011 *)

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A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

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A193095 Number of times n can be written as concatenation of exactly two nonzero squares in decimal representation.

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A038670 Concatenations of two squares in two ways.

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A193097 Numbers that are the concatenation of exactly one pair of nonzero squares.

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