A018796 -id:A018796 - cp's OEIS frontend

A272677 Number of digits in A018796 - number of digits in n.

0, 0, 1, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 0, 2, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 0, 2, 2, 1, 2, 2, 3, 2, 1, 3, 2, 2, 3, 1, 3, 0, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 2, 3

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

nonn

These are the numbers d mentioned in Robert Israel's comment in A018851.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A018851, A018796.

A018851 a(n)^2 is smallest square beginning with n.

Original entry on oeis.org

0, 1, 5, 6, 2, 23, 8, 27, 9, 3, 10, 34, 11, 37, 12, 39, 4, 42, 43, 14, 45, 46, 15, 48, 49, 5, 51, 52, 17, 54, 55, 56, 18, 58, 59, 188, 6, 61, 62, 63, 20, 203, 65, 66, 21, 213, 68, 69, 22, 7, 71, 72, 23, 73, 74, 235, 75, 24, 241, 77, 78, 247, 25, 251, 8, 81, 257, 26, 83, 263, 84, 267, 27

Offset: 0

David W. Wilson

The following discussion is based on comments from David A. Corneth, Robert Israel, N. J. A. Sloane, and Chai Wah Wu. (Start)
As the graph shows, the points belong to various "curves". For each n there is a value d = d(n) such that n*10^d <= a(n)^2 < (n+1)*10^d, and so on this curve, a(n) ~ sqrt(n)*10^(d/2). The values of d(n) are given in A272677.
For a given n, d can range from 0 (if n is a square) to d_max, where it appears that d_max approx. equals 3 + floor( log_10(n/25) ). The successive points where d_max increases are given in A272678, and that entry contains more precise conjectures about the values.
For example, in the range 2600 = A272678(5) <= n < 25317 = A272678(6), d_max is 5. This is the upper curve in the graph that is seen if the "graph" button is clicked, and on this curve a(n) is about sqrt(n)*10^(5/2). (End)

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
Zak Seidov, Blog entry
Zak Seidov, Blog entry [Cached copy, pdf format, with permission]

Cf. A018796 (the squares), A272677, A272678.

A265432 is a more complicated sequence in the same spirit.

f:= proc(n) local d,m;
  for d from 0 do
    m:= ceil(sqrt(10^d*n));
    if m^2 < 10^d*(n+1) then return m fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2015

a(n)=k=1; while(k,d=digits(k^2); D=digits(n); if(#D<=#d,c=1; for(i=1,#D,if(D[i]!=d[i],c=0;break));if(c,return(k)));k++)
vector(100,n,a(n)) \\ Derek Orr, Jul 26 2015

a(n) >= sqrt(n), for all n >= 0 with equality when n is a square. - Derek Orr, Jul 26 2015

Added initial 0. - N. J. A. Sloane, May 21 2016

Comments revised by N. J. A. Sloane, Jul 17 2016

A265432 a(n) = smallest k with concat(1,k) and concat(n,k) both square numbers.

Original entry on oeis.org

225, 6, 1025, 6, 225, 9937257544619140625, 80625, 225, 19025, 14797831640625, 5625, 89450791534674072265625, 96, 69, 44, 21, 1993672119140625, 2002541101386962890625, 225, 6, 8734765625, 99758030615478515625, 5625, 863225, 80625, 6, 40625, 225, 890625, 158764150390625

Offset: 0

David W. Wilson and Robert G. Wilson v, Dec 08 2015

k must be a positive integer (and of course cannot begin with 0). - N. J. A. Sloane, May 19 2016

Every term is a member of A272671, by definition. Certainly every term of A272671 which is a power of 100 times an earlier term of A272671 (such as 600, 2100, 4400) will not appear, by the "smallest k" condition. Does every other term of A272671 (that is, the terms of A272684) eventually appear? See A272685 and A273369 for the first appearance of these terms. - Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

			a(0) = 225 because 1225 is a square as is (0)225. (In other words, 225 is the first term in A272672 that is itself a square). - _N. J. A. Sloane_, May 21 2016
a(2) = 1025 because concat(1,1025) = 11025 = 105^2 and concat(2,1025) = 21025 = 145^2.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000, May 25 2016 [First 10000 terms from David W. Wilson, Dec 08 2015]
N. J. A. Sloane, Table of n, a(n), sqrt(concat(n,a(n))), sqrt(concat(1,a(n))) (based on _David W. Wilson_ 10000-term b-file)
David W. Wilson, Table of n, a(n) for n = 0..10000, May 25 2016 [The graph of the first 10000 terms looks rather different from the graph of the first 20000 terms, so this file is worth preserving. - _N. J. A. Sloane_, May 25 2016]

Cf. A045855, A272671, A018796, A272684, A272685 and A273369 (smallest inverse).
For records see A272674, A272675.
For square roots referred to in definition see A272682, A272683.
A018851 is a simpler sequence in the same spirit.

<< Combinatorica`
A265432[0] = 225;
A265432[1] = 6;
A265432[n_] := Block[{x = {-1, 1, 0, 1}[[Mod[n, 4, 1]]], d = Infinity, l, i}, While[d > Sqrt[10.0^(x - 1)] (Sqrt[10.0 n + 1] - Sqrt[11.0]), x++; d = Infinity; l = Divisors[((n - 1) 10^x)/4]; i = BinarySearch[l, 0.5 Sqrt[(n + 1) 10.0^x - 1] - 0.5 Sqrt[2*10.0^x - 1]]; If[i <= Length@l, d = 2*l[[i + 1/2]]]]; (((n - 1) 10^x - d^2)/(2 d))^2 - 10^x] (* Davin Park, Apr 11 2017 *)

A018797 Smallest cube that begins with n.

Original entry on oeis.org

0, 1, 27, 343, 4096, 512, 64, 729, 8, 9261, 1000, 110592, 125, 1331, 140608, 15625, 166375, 1728, 185193, 19683, 205379, 216, 226981, 238328, 24389, 250047, 262144, 27, 287496, 29791, 300763, 314432, 32768, 3375, 343, 35937, 3652264, 373248

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from T. D. Noe)

Cf. A018852, A018796.

a[n_] := For[idn = IntegerDigits[n]; k = 1, True, k++, idk = IntegerDigits[k^3]; If[Length[idn] <= Length[idk], If[idk[[1 ;; Length[idn]]] == idn, Return[k^3]]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 1000}] (* Jean-François Alcover, Mar 13 2016 *)
id[n_]:=With[{idn=IntegerDigits[n]},idn==Take[IntegerDigits[#], Length[ idn]]&]; With[{c=Range[0,250]^3},Table[SelectFirst[c,id[k]],{k,0,40}]]//Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 08 2018 *)

for n in range(1, 10**3):
  for k in range(10**3):
    if str(k**3).startswith(str(n)):
      print(k**3, end=', ')
      break
  n += 1 # Derek Orr, Aug 03 2014

a(n) = A018852(n)^3. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A030666 Smallest nontrivial extension of n which is a square.

Original entry on oeis.org

16, 25, 36, 49, 529, 64, 729, 81, 900, 100, 1156, 121, 1369, 144, 1521, 169, 1764, 1849, 196, 2025, 2116, 225, 2304, 2401, 256, 2601, 2704, 289, 2916, 3025, 3136, 324, 3364, 3481, 35344, 361, 3721, 3844, 3969, 400, 41209, 4225, 4356, 441

Offset: 1

Patrick De Geest

A trivial extension would mean appending no digits at all when n is already a square. With trivial extensions allowed, this sequence becomes A018796.

			80 is not a perfect square, but 81 = 9^2, so a(8) = 81.
Although 9 is already a square, we need to append at least one digit. However, none of 90, 91, 92, ..., 99 are squares. Then we try 900 = 30^2, so a(9) = 900.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A030667, A030686, A018796.

See A264604 for another version (first differs at a(9)).

# Program which computes 20000 terms, from N. J. A. Sloane, Nov 24 2015
for b from 1 to 20000 do
sw1:=-1:
for p from 1 to 6 do
bp:=b*10^p;
for i from 0 to 10^p-1 do
if issqr(bp+i) then lprint(b,bp+i); sw1:=1; break; fi;
                       od:
if sw1 > 0 then break; fi;
                 od:
if sw1 < 0 then lprint("failed at b = ",b); fi;
                      od:

from gmpy2 import isqrt
def A030666(n):
    d, nd = 10, 10*n
    while True:
        x = (isqrt(nd-1)+1)**2
        if x < nd+d:
            return int(x)
        d *= 10
        nd *= 10 # Chai Wah Wu, May 24 2016

a(n) = A030667(n)^2.

A272679 a(n)^2 is the smallest square whose binary expansion begins with the binary expansion of n.

Original entry on oeis.org

0, 1, 2, 5, 2, 9, 5, 11, 4, 3, 9, 19, 5, 21, 15, 11, 4, 33, 6, 25, 9, 13, 19, 39, 7, 5, 29, 21, 15, 61, 11, 45, 8, 23, 33, 67, 6, 49, 35, 25, 9, 73, 13, 53, 107, 19, 77, 39, 79, 7, 10, 81, 29, 83, 59, 21, 15, 43, 61, 87, 11, 89, 63, 45, 8, 129, 23, 93, 33, 47

Offset: 0

N. J. A. Sloane, May 22 2016

			a(10)=9, because 9^2 = 81 = 1010001_2 begins with 1010 = 10_2.

Allan C. Wechsler, posting to math-fun mailing list May 22 2016.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A007088, A018851, A018796, A272680, A272681.

from gmpy2 import isqrt
def A272679(n):
    if n == 0:
        return 0
    else:
        d, nd = 1, n
        while True:
            x = isqrt(nd-1)+1
            if x**2 < nd+d:
                return int(x)
            d *= 2
            nd *= 2 # Chai Wah Wu, May 22 2016

More terms from Chai Wah Wu, May 22 2016

A077346 Triangle in which n-th row contains n smallest squares beginning with n.

Original entry on oeis.org

1, 25, 225, 36, 324, 361, 4, 49, 400, 441, 529, 576, 5041, 5184, 5329, 64, 625, 676, 6084, 6241, 6400, 729, 784, 7056, 7225, 7396, 7569, 7744, 81, 841, 8100, 8281, 8464, 8649, 8836, 80089, 9, 900, 961, 9025, 9216, 9409, 9604, 9801, 90000

Offset: 1

Amarnath Murthy, Nov 05 2002

A018796 contains initial terms of rows. Subsidiary sequence: There can be a rearrangement of squares in groups so that the n-th group contains n squares beginning with n and not occurring earlier. E.g. the initial term of row 25 would be 256 and not 25.

			Triangle begins:
    1;
   25, 225;
   36, 324,  361;
    4,  49,  400,  441;
  529, 576, 5041, 5184, 5329;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A018796, A077347, A077348, A077349.

g:= proc(n,d)
    local i,r;
    r:= ilog10(n)+1;
    seq(i^2, i = ceil(sqrt(n*10^(d-r))) .. ceil(sqrt((n+1)*10^(d-r)))-1)
end proc:
f:= proc(n)
   local R,count,v,d;
   count:= 0; R:= NULL;
   for d from 1 + ilog10(n) do
     v:= g(n,d);
     R:= R,v;
     count:= count + nops([v]);
     if count >= n then return R[1..n] fi
   od
end proc:
f(1):= 1:
for i from 1 to 10 do f(i) od; # Robert Israel, Jan 28 2025

row(n) = my(list=List(), k=1); while (#list != n, if (strsplit(Str(k^2), Str(n))[1] == "", listput(list, k^2)); k++); Vec(list); \\ Michel Marcus, Feb 08 2023

More terms from Michel Marcus, Feb 08 2023

A077347 Final terms of rows of A077346.

Original entry on oeis.org

1, 225, 361, 441, 5329, 6400, 7744, 80089, 90000, 101124, 112896, 122500, 135424, 145924, 157609, 165649, 179776, 1806336, 1901641, 2013561, 2114116, 2220100, 2325625, 2427364, 2528100, 2637376, 2742336, 2845969, 2951524, 3059001

Offset: 1

Amarnath Murthy, Nov 05 2002

a(n) = n-th square beginning with n. A018796 contains initial terms of rows. Subsidiary sequence: There can be a rearrangement of squares in groups so that the n-th group contains n squares beginning with n and not occurring earlier. E.g., the initial term of row 25 would be 256 and not 25 and accordingly the final terms would differ.

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

Cf. A018796, A077346, A077348, A077349.

startsWith := proc(n,dig) local nshft ; nshft := n ; while nshft > dig do nshft := floor(nshft/10) ; od ; if dig = nshft then RETURN(true) ; else RETURN(false) ; fi ; end: A077347 := proc(n) local candid,c; candid := 1 ; for c from 1 to n do while not startsWith(candid^2,n) do candid := candid+1 ; od ; if c = n then RETURN(candid^2) ; fi ; candid := candid+1 ; od ; end: for n from 1 to 50 do printf("%a,",A077347(n)) ; od ; # R. J. Mathar, Nov 12 2006

More terms from R. J. Mathar, Nov 12 2006

A077349 a(n) = A077347(n)^(1/2).

Original entry on oeis.org

1, 15, 19, 21, 73, 80, 88, 283, 300, 318, 336, 350, 368, 382, 397, 407, 424, 1344, 1379, 1419, 1454, 1490, 1525, 1558, 1590, 1624, 1656, 1687, 1718, 1749, 1779, 1806, 1836, 1865, 1895, 1918, 1949, 6165, 6247, 6326, 6409, 6487, 6565, 6641, 6720, 6795, 6868, 6943, 7013, 7089

Offset: 1

Amarnath Murthy, Nov 05 2002

Andrew Howroyd, Table of n, a(n) for n = 1..500

Main diagonal of A077348.

Cf. A077346, A018796, A077347.

a(n)=row(n)[n] \\ row(n) defined in A077348. - Andrew Howroyd, Dec 11 2024

a(8) onwards from Andrew Howroyd, Dec 11 2024

A077348 a(n) = A077346(n)^(1/2).

Original entry on oeis.org

1, 5, 15, 6, 18, 19, 2, 7, 20, 21, 23, 24, 71, 72, 73, 8, 25, 26, 78, 79, 80, 27, 28, 84, 85, 86, 87, 88, 9, 29, 90, 91, 92, 93, 94, 283, 3, 30, 31, 95, 96, 97, 98, 99, 300, 10, 32, 33, 100, 101, 102, 103, 104, 317, 318, 34, 105, 106, 107, 108, 109, 332, 333, 334, 335, 336

Offset: 1

Amarnath Murthy, Nov 05 2002

			Triangle begins:
   1;
   5, 15;
   6, 18, 19;
   2,  7, 20, 21;
  23, 24, 71, 72, 73;
  ...

Cf. A077346, A018796, A077347, A077349.

row(n) = my(list=List(), k=1); while (#list != n, if (strsplit(Str(k^2), Str(n))[1] == "", listput(list, k)); k++); Vec(list); \\ Michel Marcus, Feb 08 2023

More terms from Michel Marcus, Feb 08 2023

cp's OEIS Frontend

A272677 Number of digits in A018796 - number of digits in n.

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A018851 a(n)^2 is smallest square beginning with n.

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A265432 a(n) = smallest k with concat(1,k) and concat(n,k) both square numbers.

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A018797 Smallest cube that begins with n.

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A030666 Smallest nontrivial extension of n which is a square.

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A272679 a(n)^2 is the smallest square whose binary expansion begins with the binary expansion of n.

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A077346 Triangle in which n-th row contains n smallest squares beginning with n.

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A077347 Final terms of rows of A077346.

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