A048646 -id:A048646 - cp's OEIS frontend

A048375 Numbers whose square is a concatenation of two nonzero squares.

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

Offset: 1

Patrick De Geest, Mar 15 1999

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

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

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

Cf. A039686, A048646.

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 *)

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

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

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

A048653 Numbers k such that the decimal digits of k^2 can be partitioned into two or more nonzero squares.

Original entry on oeis.org

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

Offset: 1

Felice Russo

			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.

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

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

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

(* 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 *)

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

Corrected and extended by Naohiro Nomoto, Sep 01 2001

Definition clarified by Harvey P. Dale, May 09 2021

A219542 Numbers n such that n^2 is a concatenation of 3 nonzero squares, leading zeros not allowed.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Nov 22 2012

			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.

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

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

A344045 Primes p such that the decimal digits of p^2 can be partitioned into two or more squares.

Original entry on oeis.org

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

Offset: 1

Harvey P. Dale, May 09 2021

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

			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.

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

Cf. A048646, A048653, A128783.

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]

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A048375 Numbers whose square is a concatenation of two nonzero squares.

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A048653 Numbers k such that the decimal digits of k^2 can be partitioned into two or more nonzero squares.

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A219542 Numbers n such that n^2 is a concatenation of 3 nonzero squares, leading zeros not allowed.

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Examples

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A344045 Primes p such that the decimal digits of p^2 can be partitioned into two or more squares.

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Author

Keywords

Comments

Examples

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Programs

Mathematica