A128783 -id:A128783 - cp's OEIS frontend

A019547 Squares which are a decimal concatenation of two or more squares.

49, 100, 144, 169, 361, 400, 441, 900, 1225, 1369, 1444, 1600, 1681, 1936, 2500, 3249, 3600, 4225, 4900, 6400, 8100, 9025, 9409, 10000, 10404, 11025, 11449, 11664, 12100, 12544, 14161, 14400, 14641, 15625, 16641, 16900, 19044, 19600, 22500, 25600, 28900

Offset: 1

R. Muller

0 counts as a square here.

			1369 is a term as it can be partitioned as 1, 36 and 9. 1444 is a term as it can be partitioned as 1, 4, 4, 4. Again, 100 is 1, 0, 0.

L. Widmer, Construction of Elements of the Smarandache Square-Partial-Digital Sequence, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 145-146.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), sqrt(a(n)), all possible decompositions of a(n) into squares for n = 1..10000 (excluding solutions with leading zeros)
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. A010052, A000290, A128783.

r[n_, d_] := Catch@Block[{z = Length@d, t}, z < 1 || Do[If[IntegerQ@ Sqrt@ (t = FromDigits@Take[d, i]) && t < n && r[n, Take[d, i - z]], Throw@ True], {i, z}]]; Select[Range[4, 10^4]^2, r[#, IntegerDigits@ #] &] (* Giovanni Resta, Apr 30 2013 *)

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 issquare(int(d[:i])) and ok(int(d[i:]), c+1): return True
    return False
print([r*r for r in range(180) if ok(r*r, 1)]) # Michael S. Branicky, Jul 10 2021

n^2 < a(n) < 100n^2. - Charles R Greathouse IV, Sep 19 2012

a(n) = A128783(n)^2. - Michael S. Branicky, Jul 10 2021

More terms from Christian N. K. Anderson, Apr 30 2013

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

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]

cp's OEIS Frontend

A019547 Squares which are a decimal concatenation of two or more squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica