A048646 -id:A048646

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A048375

Numbers whose square is a concatenation of two nonzero squares.

+10
7

7, 13, 19, 35, 38, 41, 57, 65, 70, 125, 130, 190, 205, 223, 253, 285, 305, 350, 380, 410, 475, 487, 570, 650, 700, 721, 905, 975, 985, 1012, 1201, 1250, 1265, 1300, 1301, 1442, 1518, 1771, 1900, 2024, 2050, 2163, 2225, 2230, 2277, 2402, 2435, 2530, 2850

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Leading zeros not allowed, trailing zeros are.

This means that, e.g., 95 is not in the sequence although 95^2 = 9025 could be seen as concatenation of 9 and 025 = 5^2. - M. F. Hasler, Jan 25 2016

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..3000

FORMULA

a(n) = sqrt(A039686(n)). - M. F. Hasler, Jan 25 2016

EXAMPLE

1771^2 = 3136441 = 3136_441 and 3136 = 56^2, 441 = 21^2.

MATHEMATICA

squareQ[n_] := IntegerQ[Sqrt[n]]; okQ[n_] := MatchQ[IntegerDigits[n^2], {a__ /; squareQ[FromDigits[{a}]], b__ /; First[{b}] > 0 && squareQ[FromDigits[{b}]]}]; Select[Range[3000], okQ] (* Jean-François Alcover, Oct 20 2011, updated Dec 13 2016 *)

PROG

(PARI) is_A048375(n)={my(p=100^valuation(n, 10)); n*=n; while(n>p*=10, issquare(n%p)&&issquare(n\p)&&n%p*10>=p&&return(1))} \\ M. F. Hasler, Jan 25 2016

(Python)

from math import isqrt

def issquare(n): return isqrt(n)**2 == n

def ok(n):

d = str(n)

for i in range(1, len(d)):

if d[i] != '0' and issquare(int(d[:i])) and issquare(int(d[i:])):

return True

return False

print([r for r in range(2851) if ok(r*r)]) # Michael S. Branicky, Jul 13 2021

CROSSREFS

Cf. A039686, A048646.

KEYWORD

nonn,easy,nice,base

AUTHOR

Patrick De Geest, Mar 15 1999

STATUS

approved

A048653

Numbers k such that the decimal digits of k^2 can be partitioned into two or more nonzero squares.

+10
4

7, 12, 13, 19, 21, 35, 37, 38, 41, 44, 57, 65, 70, 107, 108, 112, 119, 120, 121, 125, 129, 130, 190, 191, 204, 205, 209, 210, 212, 223, 253, 285, 305, 306, 315, 342, 343, 345, 350, 369, 370, 379, 380, 408, 410, 413, 440, 441, 475, 487, 501, 538, 570, 642, 650

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

12 is present because 12^2=144 can be partitioned into three squares 1, 4 and 4.

108^2 = 11664 = 1_16_64, 120^2 = 14400 = 1_4_400, so 108 and 120 are in the sequence.

MATHEMATICA

(* This non-optimized program is not suitable to compute a large number of terms. *) split[digits_, pos_] := Module[{pos2}, pos2 = Transpose[{Join[ {1}, Most[pos+1]], pos}]; FromDigits[Take[digits, {#[[1]], #[[2]]}]]& /@ pos2]; sel[n_] := Module[{digits, ip, ip2, accu, nn}, digits = IntegerDigits[n^2]; ip = IntegerPartitions[Length[digits]]; ip2 = Flatten[ Permutations /@ ip, 1]; accu = Accumulate /@ ip2; nn = split[ digits, #]& /@ accu; SelectFirst[nn, Length[#]>1 && Flatten[ IntegerDigits[#] ] == digits && AllTrue[#, #>0 && IntegerQ[Sqrt[#]]&]&] ]; k = 1; Reap[Do[If[(s = sel[n]) != {}, Print["a(", k++, ") = ", n, " ", n^2, " ", s]; Sow[n]], {n, 1, 10^4}]][[2, 1]] (* Jean-François Alcover, Sep 28 2016 *)

PROG

(Haskell)

a048653 n = a048653_list !! (n-1)

a048653_list = filter (f . show . (^ 2)) [1..] where

f zs = g (init $ tail $ inits zs) (tail $ init $ tails zs)

g (xs:xss) (ys:yss)

| h xs = h ys || f ys || g xss yss

| otherwise = g xss yss

where h ds = head ds /= '0' && a010052 (read ds) == 1

g _ _ = False

-- Reinhard Zumkeller, Oct 11 2011

(Python)

from math import isqrt

def issquare(n): return isqrt(n)**2 == n

def ok(n, c):

if n%10 in {2, 3, 7, 8}: return False

if issquare(n) and c > 1: return True

d = str(n)

for i in range(1, len(d)):

if d[i] != '0' and issquare(int(d[:i])) and ok(int(d[i:]), c+1):

return True

return False

def aupto(lim): return [r for r in range(lim+1) if ok(r*r, 1)]

print(aupto(650)) # Michael S. Branicky, Jul 10 2021

CROSSREFS

Cf. A048646, A048375, A010052, A000290; subsequence of A128783.

KEYWORD

base,nice,nonn

AUTHOR

Felice Russo

EXTENSIONS

Corrected and extended by Naohiro Nomoto, Sep 01 2001

Definition clarified by Harvey P. Dale, May 09 2021

STATUS

approved

A219542

Numbers n such that n^2 is a concatenation of 3 nonzero squares, leading zeros not allowed.

+10
1

12, 21, 37, 44, 107, 108, 120, 129, 191, 204, 209, 210, 223, 306, 315, 342, 343, 345, 370, 408, 413, 440, 501, 642, 696, 804, 955, 959, 982, 995, 1002, 1044, 1063, 1065, 1070, 1080, 1107, 1169, 1200, 1275, 1281, 1290, 1301, 1306, 1315, 1349, 1385, 1503, 1910

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Zak Seidov, Table of n, a(n) for n = 1..1500

EXAMPLE

a(1) = 12: 12^2 = 144, 1 = 1^2, 4 = 2^2, 4 = 2^2;

a(1500) = 3176900^2 = 100, 9, 2693610000, 100 = 10^2, 9 = 3^2, 2693610000 = 51900^2.

CROSSREFS

Cf. A039686, A048375, A048646, A145848, A147608.

KEYWORD

nonn,base

AUTHOR

Zak Seidov, Nov 22 2012

STATUS

approved

A344045

Primes p such that the decimal digits of p^2 can be partitioned into two or more squares.

+10
1

7, 13, 19, 37, 41, 97, 107, 191, 223, 379, 397, 487, 509, 701, 997, 1049, 1063, 1093, 1201, 1301, 1801, 1907, 2011, 2029, 3019, 3169, 3319, 3371, 3767, 4013, 4451, 5009, 5011, 5081, 5099, 5693, 6037, 6397, 7001, 8009, 9041, 9521, 9619, 9721, 9907, 10007

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Similar to A048646 except that here zeros are permitted as squares.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..200

EXAMPLE

97 is a term because 97 is a prime and 97^2 = 9409 which can be partitioned into 9, 4, 0, and 9, each of which is a square.

MATHEMATICA

tmsQ[n_]:=Total[Boole[AllTrue[Sqrt[#], IntegerQ]&/@Rest[Table[FromDigits/@ TakeList[IntegerDigits[n^2], q], {q, Flatten[Permutations/@ IntegerPartitions[ IntegerLength[ n^2]], 1]}]]]]>0; Select[Prime[ Range[ 3000]], tmsQ]

CROSSREFS

Cf. A048646, A048653, A128783.

KEYWORD

nonn,base

AUTHOR

Harvey P. Dale, May 09 2021

STATUS

approved

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