A039686 -id:A039686 - cp's OEIS frontend

A038628 Erroneous version of A039686.

49, 169, 1225, 1444, 1681, 3249, 4225, 4900, 15625, 16900, 42025, 49729, 64009

Offset: 1

dead

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 *)

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

Original entry on oeis.org

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

A147608 Squares which are concatenation of two positive squares with possible intervening zeros.

Original entry on oeis.org

49, 169, 361, 1225, 1444, 1681, 3249, 4225, 4900, 9025, 15625, 16900, 36100, 42025, 49729, 64009, 81225, 93025, 105625, 122500, 144400, 168100, 225625, 237169, 256036, 324900, 422500, 490000, 519841, 576081, 819025, 902500, 950625, 970225

Offset: 1

Zak Seidov, Nov 08 2008

A145848 is a subsequence.

			324 = 18^2, 9 = 3^2, 3249 = 324:9 = 57^2.
9 = 3^2, 25 = 5^2, 9025 = 9:0:25 = 95^2 (intervening zero).

David W. Wilson, Table of n, a(n) for n = 1..1000

Cf. A039686 (variant with no intervening zeros).

Cf. A145848 (squares with an even number of digits, where the first half is a square and the second half is a nonzero square).

s={}; Do[n2=n*n; id=IntegerDigits[n2]; Le=Length[id]; Do[If[IntegerQ/@Sqrt[FromDigits/@{Take[id,k], Drop[id,k]}] == {True,True}, If[FromDigits@Drop[id,k]>0, Print[n2]; AppendTo[s,n2]; Break[]]], {k,1,Le-1}], {n,4,10^4}]

Edited by Charles R Greathouse IV, Apr 24 2010

Definition corrected by David W. Wilson, Nov 22 2012

A054738 Squares which can be split into two nonzero squares (perhaps with leading zeros) in exactly two different ways.

Original entry on oeis.org

64009, 1600225, 4950625, 6400900, 40005625, 160022500, 324900625, 360050625, 495062500, 624100324, 640090000, 1000140625, 4000562500, 9001265625, 16002250000, 25003515625, 32490062500, 36005062500, 49006890625, 49506250000, 62410032400, 64009000000

Offset: 0

Henry Bottomley, Apr 26 2000

			4950625=2225^2 can be split into 49=7^2 and 50625=225^2 or into 4=2^2 and 950625=975^2; 6400900=2530^2 can be split into 6400=80^2 and 900=30^2 or into 64=8^2 and 00900=30^2

Cf. A039686.

a(n) = A054737(n)^2. - Sean A. Irvine, Feb 20 2022

More terms from Sean A. Irvine, Feb 20 2022

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.

A258060 Squares, without multiplicity, that are the concatenation of two integers (without leading zeros) the product of which is also a square.

Original entry on oeis.org

49, 169, 361, 1225, 1444, 1681, 3249, 4225, 4900, 15625, 16900, 36100, 42025, 49729, 64009, 81225, 93025, 122500, 142129, 144400, 168100, 225625, 237169, 324900, 414736, 422500, 490000, 519841, 819025, 950625, 970225, 1024144, 1442401, 1562500, 1600225, 1690000, 1692601, 2079364, 2304324

Offset: 1

Reiner Moewald, Jul 26 2015

Squares that can be split up in more than one way, e.g., 4950625 with sqrt(4 * 950625) = 1950 and sqrt(49 * 50625) = 1575, appear only once.

Squares that are members of this sequence in more than one way: 4950625, 495062500, 49506250000, 4950625000000, ..., . - Robert G. Wilson v, Aug 14 2015

			169 = 13^2 can be split up into 16 and 9 and 16*9 = 144, a square.

Robert G. Wilson v, Table of n, a(n) for n = 1..1386 (first 200 terms from Reiner Moewald)

Subsequence of A039686.

p:= proc(k,n) local t; t:= n mod 10^k; t >= 10^(k-1) and issqr(t*(n-t)/10^k) end proc:
filter:= n -> ormap(p, [$1..ilog10(n)], n):
select(filter, [seq(i^2, i=1..10^4)]); # Robert Israel, Sep 22 2015

f[n_] := Block[{idn = IntegerDigits@ n, c = 0, k = 1, lmt = Floor[1 + Log10@ n]}, While[k < lmt, m = Mod[n, 10^(lmt - k)]; If[ IntegerQ@ Sqrt[ FromDigits[ Take[idn, {1, k}]] m] && m > 0 && IntegerDigits[m] == Take[idn, {k + 1, -1}], c++]; k++]; c]; Select[ Range[1700]^2, f@# > 0 &] (* Robert G. Wilson v, Aug 13 2015 *)

isok(n) = {if (issquare(n), len = #Str(n); for (k=1, len-1, na = n\10^k; nb = n%10^k; if (na && nb && (eval(Str(na,nb))==n) && issquare(na*nb), return (1));););} \\ Michel Marcus, Oct 09 2015

import math
list =[]
for i in range(1,100000):
   a = i*i
   b = str(a)
   l = len(b)
   for j in range(1, l):
      a_1 = b[:j]
      a_2 = b[j:]
      c = int(a_1)* int(a_2)
      sqrt_c = int(math.sqrt(int(c)))
      if (sqrt_c * sqrt_c == c) and (int(a_2[:1]) > 0):
         if not a in list:
            list.append(a)
         list.append(a)
print(list)

A351925 Squares which are the concatenation of two primes.

Original entry on oeis.org

25, 289, 361, 529, 729, 2401, 2601, 2809, 4761, 5329, 5929, 7569, 11449, 11881, 15129, 19881, 21609, 22801, 23409, 24649, 25281, 26569, 29241, 29929, 31329, 34969, 36481, 39601, 47961, 52441, 53361, 54289, 57121, 58081, 59049, 71289, 77841, 83521, 89401

Offset: 1

Max Z. Scialabba, Feb 25 2022

The first term that is the concatenation of two primes in more than one way is a(11) = 5929 = 5 | 929 = 59 | 29. - Robert Israel, Oct 01 2023

			25 is the concatenation of 2 and 5, both primes.
4761 is the concatenation of 47 and 61, both primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290 (squares), A039686, A106582, inverse of A167535.

Cf. A038692, A225135.

L:= NULL: count:=0:
for x from 1 by 2 while count < 100 do
  xs:= x^2;
  for i from 1 to ilog10(xs) do
    a:= xs mod 10^i;
    if a > 10^(i-1) and isprime(a) then
      b:= (xs-a)/10^i;
      if isprime(b) then
        L:= L, xs; count:= count+1; break
      fi fi
od od:
L; # Robert Israel, Oct 01 2023

isb(n)={my(d=10); while(dAndrew Howroyd, Feb 26 2022

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        s = str(k*k)
        if any(s[i] != '0' and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s))):
            yield k*k
print(list(islice(agen(), 39))) # Michael S. Branicky, Feb 26 2022

Intersection of A106582 and A000290.

cp's OEIS Frontend

A038628 Erroneous version of A039686.

Views

Author

Keywords

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A147608 Squares which are concatenation of two positive squares with possible intervening zeros.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A054738 Squares which can be split into two nonzero squares (perhaps with leading zeros) in exactly two different ways.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

A258060 Squares, without multiplicity, that are the concatenation of two integers (without leading zeros) the product of which is also a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A351925 Squares which are the concatenation of two primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Python