A265432 -id:A265432 - cp's OEIS frontend

A272685 a(n) = smallest m such that A265432(m) = A272684(n), or -1 if no such m exists.

1, 15, 14, 13, 12, 217, 0, 215, 45, 213, 44, 43, 209, 42, 207, 2, 573, 1327, 572, 130, 185, 570, 1492, 569, 78, 568, 128, 567, 1318, 1498, 565, 188, 564, 10, 563, 1312, 562, 1504, 1309, 560, 1507, 693, 74, 558, 1510, 557, 192, 2360, 6085, 350, 1480, 708, 6083, 3643

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 24 2016

It is conjectured that all terms of A272684 eventually appear in A265432.

See A273369 for a version that ignores the fact that terms in A272671 ending in 0 cannot appear in A265432.

			A272684(1) = 6 first appears in A265432 at index 1, so a(1) = 1.
A272684(7) = 225 first appears in A265432 at index 0, so a(7) = 0.

Cf. A272684, A272671, A265432.

See A273369 for another version.

A272674 Records in A265432.

Original entry on oeis.org

225, 1025, 9937257544619140625, 89450791534674072265625, 13268252282260894775390625, 1110239752327213382720947265625, 23880648486131856540581070818006992340087890625, 82567607657239917717538563557900488376617431640625

Offset: 1

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

nonn

N. J. A. Sloane and Chai Wah Wu, Table of n, a(n) for n = 1..112 a(n) for n = 1..96 from N. J. A. Sloane.

Cf. A265432, A272675.

A272675 Indices where records occur in A265432.

Original entry on oeis.org

0, 2, 5, 11, 30, 31, 32, 98, 101, 1092, 1116, 1143, 1250, 1251, 1601, 1715, 1757, 2712, 2714, 2771, 2778, 2781, 2844, 2981, 4398, 5298, 5421, 5423, 9189, 9191, 9195, 9200, 9204, 9210, 9407, 9411, 9414, 9420, 9422, 9432, 9434, 9438, 9440, 9630, 9632, 9644, 9650, 9656, 9662, 9666, 9671

Offset: 1

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

nonn

N. J. A. Sloane and Chai Wah Wu, Table of n, a(n) for n = 1..112 a(n) for n = 1..96 from N. J. A. Sloane

Cf. A265432, A272674.

A272682 Square root of concatenation of n and A265432(n).

Original entry on oeis.org

15, 4, 145, 6, 65, 7741915625, 825, 85, 905, 30245625, 325, 1090619453125, 36, 37, 38, 39, 402484375, 414731890625, 135, 14, 456875, 46901578125, 475, 4885, 1575, 16, 1625, 165, 5375, 170759375, 54893244140625, 17638317373046875, 1795516819326995849609375, 57642802733386962890625, 185

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

The square of a(n) is one of the two squares that are required by the definition of A265432(n). The other is in A272683.

Of course if it ever happens that A265432(n) = -1, the definition will need to be changed.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, A265432(n), sqrt(concat(n,A265432(n))), sqrt(concat(1,A265432(n))) (based on _David W. Wilson_ 10000-term b-file for A265432)

Cf. A265432, A272683.

A272683 Square root of concatenation of 1 and A265432(n).

Original entry on oeis.org

35, 4, 105, 4, 35, 4465115625, 425, 35, 345, 10714375, 125, 435259453125, 14, 13, 12, 11, 109515625, 109556109375, 35, 4, 136875, 14133578125, 125, 1365, 425, 4, 375, 35, 1375, 34040625, 10642755859375, 3333202626953125, 351966828673004150390625, 11076674002613037109375, 35, 351172546875, 33875, 11, 12, 13

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

The square of a(n) is one of the two squares that are required by the definition of A265432(n). The other is in A272682.

Of course if it ever happens that A265432(n) = -1, the definition will need to be changed.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, A265432(n), sqrt(concat(n,A265432(n))), sqrt(concat(1,A265432(n))) (based on _David W. Wilson_ 10000-term b-file for A265432)

Cf. A265432, A272682.

A273369 a(n) is the smallest m such that A265432(m) = A272671(n), or -1 if no such m exists.

Original entry on oeis.org

1, 15, 14, 13, 12, 217, 0, 215, 45, 213, 44, -1, 43, 209, 42, 207, 2, 573, 1327, 572, 130, -1, 185, 570, 1492, 569, 78, 568, 128, 567, 1318, -1, 1498, 565, 188, 564, 10, 563, 1312, 562, 1504, -1, 1309, 560, 1507, 693, 74, 558, 1510, 557, 192

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

sign

Every entry in A265432 appears in A272671.
a(n) = -1 whenever A272671(n) ends in 0, because every such entry ends in 00 and anytime concat(1,k,00) and concat(n,k,00) are both perfect squares, concat(1,k) and concat(n,k) are also both perfect squares.
Does every term in A272671 that does not end in 0 appear in A265432?
See A272685 for a version that takes into account the fact that terms in A272671 ending in 0 cannot appear in A265432.

			A272671(6) = 156. A265432(217) = 156, but A265432(m) does not equal 156 for any m < 217. So a(6) = 217.

Nathan Fox, Table of n, a(n) for n = 1..62

Cf. A265432, A272671.

See A272685 for another version.

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

A272671 Numbers k such that the decimal number 1k is a square.

Original entry on oeis.org

6, 21, 44, 69, 96, 156, 225, 296, 369, 444, 521, 600, 681, 764, 849, 936, 1025, 1236, 1449, 1664, 1881, 2100, 2321, 2544, 2769, 2996, 3225, 3456, 3689, 3924, 4161, 4400, 4641, 4884, 5129, 5376, 5625, 5876, 6129, 6384, 6641, 6900, 7161, 7424, 7689, 7956, 8225, 8496, 8769, 9044, 9321, 9600

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

			44 is a member because 144 = 12^2 is a square.
0 is not a member because 10 is not a square.

Davin Park, Table of n, a(n) for n = 1..20000

Cf. A265432, A272672, A045855 (squares beginning with 1), A272684, A272685.

[n: n in [1..10000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(1)))]; // Marius A. Burtea, Mar 21 2019

t1:=[];
for k from 1 to 20000 do
if issqr(k+10^length(k)) then t1:=[op(t1),k]; fi;
od;
t1;

Flatten[n /. Solve[10^# + n == a^2 && 10^(# - 1) <= n < 10^# && a > 0, {n, a}, Integers] & /@ Range[3]] (* Davin Park, Feb 05 2017 *)
Select[Range[10000],IntegerQ[Sqrt[10^IntegerLength[#]+#]]&] (* Harvey P. Dale, Jul 20 2025 *)

isok(n) = issquare(eval(concat(1, Str(n)))); \\ Michel Marcus, Mar 21 2019

from sympy.ntheory.primetest import is_square
def ok(n): return is_square(int('1'+str(n)))
print(list(filter(ok, range(9601)))) # Michael S. Branicky, Jun 21 2021

Extended by Davin Park, Feb 05 2017

A272672 Numbers n such that the decimal concatenations 1n and 2n are both squares.

Original entry on oeis.org

1025, 102500, 1390625, 10250000, 96700625, 139062500, 1025000000, 9670062500, 13906250000, 102500000000, 967006250000, 1390625000000, 10250000000000, 17654697265625, 96700625000000, 139062500000000, 910400191015625

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

The sequence is infinite because all the numbers 1025*100^k are members.
It would be nice to have the subsequence of "primitive" terms, those that do not end in an even number of zeros.
Let v be a number such that v^2 starts with 1. Let w^2 have the same digits as v^2 except that the initial digit is a 2. Then (w + v) * (w - v) = w^2 - v^2 = 10^m for some integer m. For the  "primitive" terms, w + v turns out to be 250, 8000, 31250 etc. w - v turns out to be 40, 1250, 3200 etc. Given such w + v and w - v it is easy to find primitive elements. Furthermore, v must lie in (sqrt(11), sqrt(20)) * sqrt(10)^i and w must lie in (sqrt(21), sqrt(30)) * sqrt(10)^i for some integer i. - David A. Corneth, May 20 2016

			1025 is a member because 11025 = 105^2 and 21025 = 145^2.

Nathan Fox, Table of n, a(n) for n = 1..728
Chai Wah Wu, Primitive terms < 10^1000

Cf. A265432, A272671.

[n: n in [1..10000000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(1))) and IsSquare(Seqint(Intseq(n) cat Intseq(2)))]; // Marius A. Burtea, Mar 21 2019

t1:=[];
for k from 1 to 2000000 do
if issqr(k+10^length(k)) and
issqr(k+2*10^length(k)) then t1:=[op(t1),k]; fi;
od;
t1;

is(n)=issquare(eval(Str(1,n))) && issquare(eval(Str(2,n))) \\ Charles R Greathouse IV, May 20 2016

a(5)-a(17) from Alois P. Heinz, May 20 2016

A272684 Terms of A272671 which are not a power of 100 times an earlier term of A272671.

Original entry on oeis.org

6, 21, 44, 69, 96, 156, 225, 296, 369, 444, 521, 681, 764, 849, 936, 1025, 1236, 1449, 1664, 1881, 2321, 2544, 2769, 2996, 3225, 3456, 3689, 3924, 4161, 4641, 4884, 5129, 5376, 5625, 5876, 6129, 6384, 6641, 7161, 7424, 7689, 7956, 8225, 8496, 8769, 9044, 9321, 9881

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 24 2016

These are the only terms in A272671 that are candidates for terms of A265432.

Cf. A272671, A265432, A272685.

cp's OEIS Frontend

A272685 a(n) = smallest m such that A265432(m) = A272684(n), or -1 if no such m exists.

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A272674 Records in A265432.

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A272675 Indices where records occur in A265432.

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A272682 Square root of concatenation of n and A265432(n).

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A272683 Square root of concatenation of 1 and A265432(n).

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A273369 a(n) is the smallest m such that A265432(m) = A272671(n), or -1 if no such m exists.

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

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A272671 Numbers k such that the decimal number 1k is a square.

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A272672 Numbers n such that the decimal concatenations 1n and 2n are both squares.

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Magma

Maple

PARI