A053061 -id:A053061 - cp's OEIS frontend

A061082 a(n) = A053061(n)/n.

11, 12, 13, 104, 105, 106, 107, 108, 109, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 10032, 10033, 10034, 10035, 10036, 10037, 10038, 10039, 10040, 10041, 10042

Offset: 1

Amarnath Murthy, Apr 17 2001

A. Murthy, A note on the conjecture that the Smarandache nn-square (n concatenated with n square) sequence contains no perfect squares (to be published in the Smarandache Notions Journal).

a(n) = 10^floor(1 + 2log_10 n) + n. - Charles R Greathouse IV, Sep 20 2012

More terms from David Wasserman, Oct 02 2005

A055436 a(n) = concatenation of n^2 and n.

Original entry on oeis.org

11, 42, 93, 164, 255, 366, 497, 648, 819, 10010, 12111, 14412, 16913, 19614, 22515, 25616, 28917, 32418, 36119, 40020, 44121, 48422, 52923, 57624, 62525, 67626, 72927, 78428, 84129, 90030, 96131, 102432, 108933, 115634, 122535, 129636, 136937, 144438, 152139

Offset: 1

Henry Bottomley, May 18 2000

Cf. A053061, A055437 (10n^2+n), A055438 (100n^2+n).

[Seqint(Intseq(n) cat Intseq(n^2)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015

a:= n-> parse(cat(n*n, n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 03 2015

Table[n^2*10^IntegerLength[n] + n, {n, 36}] (* Jayanta Basu, Jul 12 2013 *)
Table[FromDigits[Join[IntegerDigits[n^2], IntegerDigits[n]]],{n, 40}] (* Vincenzo Librandi, Jan 03 2015 *)

a(n) = n^2*floor(log_10(n) + 1) + n.

a(n) = A055437(n) if 1 <= n < 10, a(n) = A055438(n) if 10 <= n < 100.

A061086 a(n) is the concatenation of n with n^3.

Original entry on oeis.org

11, 28, 327, 464, 5125, 6216, 7343, 8512, 9729, 101000, 111331, 121728, 132197, 142744, 153375, 164096, 174913, 185832, 196859, 208000, 219261, 2210648, 2312167, 2413824, 2515625, 2617576, 2719683, 2821952, 2924389, 3027000, 3129791, 3232768, 3335937, 3439304

Offset: 1

Amarnath Murthy, Apr 19 2001

			a(13) = 132197, where 2197 = 13^3.

Felice Russo, A set of Smarandache Functions, sequences and conjectures in number theory, page 65.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A020338, A053061.

[Seqint(Intseq(n^3) cat Intseq(n)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015

Table[FromDigits[Join[IntegerDigits[n], IntegerDigits[n^3]]],{n, 40}] (* Vincenzo Librandi, Jan 03 2015 *)

def a(n): return int(str(n) + str(n**3))
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Nov 28 2021

Offset corrected by Charles R Greathouse IV, Sep 20 2012

More terms from Vincenzo Librandi, Jan 03 2015

A239459 Concatenation of n^3 and n.

Original entry on oeis.org

11, 82, 273, 644, 1255, 2166, 3437, 5128, 7299, 100010, 133111, 172812, 219713, 274414, 337515, 409616, 491317, 583218, 685919, 800020, 926121, 1064822, 1216723, 1382424, 1562525, 1757626, 1968327, 2195228, 2438929, 2700030, 2979131, 3276832, 3593733

Offset: 1

Colin Barker, Mar 19 2014

a(n) is divisible by n.

			a(13) = 219713 because 2197 = 13^3 and 13 = 13^1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053061, A055436, A061086, A239460, A239461.

Cf. A061087, A239462, A239463, A239464.

Table[ToExpression[ToString[n^3]<>ToString[n]],{n,1,30}] (* Vaclav Kotesovec, Mar 24 2014 *)
Table[n^3*10^IntegerLength[n]+n,{n,40}] (* Harvey P. Dale, Sep 07 2020 *)

PARI
```
vector(100, n, eval(Str(n^3, n)))
```

A239460 Concatenation of n^2 and n^3.

Original entry on oeis.org

11, 48, 927, 1664, 25125, 36216, 49343, 64512, 81729, 1001000, 1211331, 1441728, 1692197, 1962744, 2253375, 2564096, 2894913, 3245832, 3616859, 4008000, 4419261, 48410648, 52912167, 57613824, 62515625, 67617576, 72919683, 78421952, 84124389, 90027000

Offset: 1

Colin Barker, Mar 19 2014

a(n) is divisible by n^2.

			a(9) = 81729 because 81 = 9^2 and 729 = 9^3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053061, A055436, A061086, A061087, A239459, A239461, A239462, A239463, A239464.

Table[ToExpression[ToString[n^2]<>ToString[n^3]],{n,1,30}] (* Vaclav Kotesovec, Mar 24 2014 *)
Table[FromDigits[Join[Flatten[IntegerDigits[{n^2,n^3}]]]],{n,30}] (* Harvey P. Dale, Oct 27 2019 *)

PARI
```
vector(100, n, eval(Str(n^2, n^3)))
```

A239461 Concatenation of n^3 and n^2.

Original entry on oeis.org

11, 84, 279, 6416, 12525, 21636, 34349, 51264, 72981, 1000100, 1331121, 1728144, 2197169, 2744196, 3375225, 4096256, 4913289, 5832324, 6859361, 8000400, 9261441, 10648484, 12167529, 13824576, 15625625, 17576676, 19683729, 21952784, 24389841, 27000900

Offset: 1

Colin Barker, Mar 19 2014

a(n) is divisible by n^2.

			a(9) = 72981 because 729 = 9^3 and 81 = 9^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053061, A055436, A061086, A239459, A239460.

Cf. A061087, A239462, A239463, A239464.

Table[ToExpression[ToString[n^3]<>ToString[n^2]],{n,1,30}] (* Vaclav Kotesovec, Mar 24 2014 *)
Table[FromDigits[Flatten[IntegerDigits/@{n^3,n^2}]],{n,30}] (* Harvey P. Dale, Nov 28 2024 *)

PARI
```
vector(100, n, eval(Str(n^3, n^2)))
```

A261713 Natural numbers that can be split into two squares, leading zeros allowed.

Original entry on oeis.org

10, 11, 14, 19, 40, 41, 44, 49, 90, 91, 94, 99, 100, 101, 104, 109, 116, 125, 136, 149, 160, 161, 164, 169, 181, 250, 251, 254, 259, 360, 361, 364, 369, 400, 401, 404, 409, 416, 425, 436, 449, 464, 481, 490, 491, 494, 499, 640, 641, 644, 649, 810, 811, 814

Offset: 1

Giovanni Teofilatto, Aug 29 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A046030, A053061, A168042, A191933.

a:= proc(n) option remember; local d, i, k;
      for k from 1+`if`(n=1, 1, a(n-1))
      do for i to length(k)-1 do
           if issqr(iquo(k, 10^i, 'd')) and
              issqr(d) then return k fi
         od
      od
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 29 2015

qQ[n_] := Floor[ Sqrt@ n]^2 == n; ok[n_] := Catch[ Do[ If[ qQ@ Floor[n / 10^k] && qQ@ Mod[n, 10^k],Throw@ True], {k, IntegerLength[n] -1}]; False]; Select[Range@ 1000, ok] (* Giovanni Resta, Aug 29 2015 *)

Name clarified by Zak Seidov, Aug 29 2015

A274620 If n^2 has an even number of digits, write n after the left half of the digits of n^2 and before the right half, otherwise if n^2 has 2t+1 digits, write n after the first t digits of n^2 and before the last t+1 digits.

Original entry on oeis.org

11, 24, 39, 146, 255, 366, 479, 684, 891, 11000, 11121, 11244, 11369, 11496, 21525, 21656, 21789, 31824, 31961, 42000, 42141, 42284, 52329, 52476, 62525, 62676, 72729, 72884, 82941, 93000, 93161, 103224, 103389, 113456, 123525, 123696, 133769, 143844, 153921, 164000

Offset: 1

N. J. A. Sloane, Jul 03 2016

In short, write n in the middle of n^2.

Portions of this sequence are sometimes given as puzzles.

			4^2 = 16 so a(4) = 1.4.6 = 146.
19^2 = 361 so a(19) = 3.19.61 = 31961.

J. A. Reeds, Personal communication to N. J. A. Sloane, Jun 04 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A053061, A055436.

nterms=100;Table[FromDigits[Flatten[Insert[d=IntegerDigits[n^2],IntegerDigits[n],Floor[Length[d]/2]+1]]],{n,nterms}] (* Paolo Xausa, Nov 24 2021 *)

def a(n):
    ss = str(n*n)
    t = len(ss)//2
    return int(ss[:t] + str(n) + ss[t:])
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Nov 24 2021

A378048 Numbers k such that k and k^2 together use at most 4 distinct decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35, 38, 40, 41, 45, 46, 50, 55, 56, 60, 63, 64, 65, 66, 68, 70, 74, 75, 76, 77, 80, 81, 83, 85, 88, 90, 91, 95, 96, 97, 99, 100, 101, 102, 105, 109, 110

Offset: 1

Jovan Radenkovicc, Nov 15 2024

Problem: Is there a real constant c such that a(n) < n^c for all positive integers n?

All of A136808, A136809, A136816, ..., A137079 are subsequences. Many if not most terms of A058411, A058413, ... ("tridigital solutions") are also in this sequence; see also Hisanori Mishima's web page for some nontrivial solutions. - M. F. Hasler, Feb 02 2025

			816 is in the sequence since 816^2 = 665856 and both together use at most 4 distinct digits.
149 is not in the sequence since 149^2 = 22201 and both together use 5 distinct digits.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 748 terms from Jovan Radenkovicc)
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020
Hisanori Mishima, Sporadic tridigital solutions
OEIS index for squares having only given digits

Supersequence of A136808, A136809, A136816, A136822.

Cf. A043537, A053061, A055436, A154155.

[n: n in [0..1000000] | #Set(Intseq(n)) le 4 and #Set(Intseq(n) cat Intseq(n^2)) le 4];

Select[Range[0, 110], Length[Union @@ IntegerDigits@ {#, #^2}] < 5 &] (* Amiram Eldar, Nov 15 2024 *)

isok(k) = #Set(concat(digits(k), digits(k^2))) <= 4; \\ Michel Marcus, Nov 15 2024

is(n)=my(s=Set(digits(n))); #s<5 && #setunion(Set(digits(n^2)),s)<5 \\ Charles R Greathouse IV, Jan 30 2025

is1(n)=#setunion(Set(digits(n^2)),Set(digits(n)))<5
ok(m)=my(d=concat(apply(k->digits(lift(k)), [m,m^2]))
test(d)=my(v=List(),D=10^d); for(n=0,D-1, if(ok(Mod(n,D)), listput(v,n))); Vec(v)
res=test(8); \\ build a list of residues mod 10^8
D=diff(concat(res,res[1]+10^8)); #D
u=List(); for(n=0,10^7, if(is1(n) && !setsearch(n,res), listput(u,n))); \\ build exceptions
setminus(select(is1,[0..n]),list(n))
list(lim)=my(v=List(u)); forstep(n=0,lim,D, if(is1(n), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jan 30 2025

def ok(n): return len(set(str(n)+str(n**2))) <= 4
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Nov 18 2024

A043537(A053061(a(n))) <= 4.

Trivially, a(n) >> n^1.66... where the exponent is log(10)/log(4) (A154155). - Charles R Greathouse IV, Jan 30 2025

A055435 (1/18)*Difference between concatenation of n and n^2 and concatenation of n^2 and n.

Original entry on oeis.org

0, -1, -3, 14, 15, 15, 14, 12, 9, 5, -55, -126, -208, -301, -405, -520, -646, -783, -931, -1090, -1260, -1441, -1633, -1836, -2050, -2275, -2511, -2758, -3016, -3285, -3565, 12144, 12342, 12529, 12705, 12870, 13024, 13167, 13299, 13420

Offset: 1

Henry Bottomley, May 18 2000

Table[((n*10^IntegerLength[x = n^2] + x) - (n^2*10^IntegerLength[n] + n))/18, {n, 40}] (* Jayanta Basu, Jul 12 2013 *)

a(n) = (A053061(n) - A055436(n))/18 = n*(10^floor(2*log_10(W2)+1) - 1 - n*(10^floor(log_10(W2)+1)-1))/18.

cp's OEIS Frontend

A061082 a(n) = A053061(n)/n.

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A055436 a(n) = concatenation of n^2 and n.

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A061086 a(n) is the concatenation of n with n^3.

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A239459 Concatenation of n^3 and n.

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A239460 Concatenation of n^2 and n^3.

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A239461 Concatenation of n^3 and n^2.

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A261713 Natural numbers that can be split into two squares, leading zeros allowed.

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Maple

Mathematica

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A274620 If n^2 has an even number of digits, write n after the left half of the digits of n^2 and before the right half, otherwise if n^2 has 2t+1 digits, write n after the first t digits of n^2 and before the last t+1 digits.

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