A272671 -id:A272671 - cp's OEIS frontend

A273364 Numbers n such that the decimal number concat(9,n) is a square.

61, 216, 409, 604, 801, 1204, 1809, 2416, 3025, 3636, 4249, 4864, 5481, 6100, 6721, 7344, 7969, 8596, 9225, 9856, 10116, 12025, 13936, 15849, 17764, 19681, 21600, 23521, 25444, 27369, 29296, 31225, 33156, 35089, 37024, 38961, 40900, 42841, 44784

Offset: 1

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

Elements are squares of integers in (sqrt(91), 10) * sqrt(10)^k without the leading 9 elements for nonnegative k. - David A. Corneth, May 20 2016

			61 is a member because 961 = 31^2 is a square.
0 is not a member because 90 is not a square.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A272671, A273357, A273358, A273359, A273360, A273361, A273362, A273363.

[n: n in [1..50000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(9)))]; // Vincenzo Librandi, Feb 20 2020

t1:=[];
for k from 1 to 50000 do
if issqr(k+9*10^length(k)) then t1:=[op(t1), k]; fi;
od;
t1;

Select[Range[45000],IntegerQ[Sqrt[9*10^IntegerLength[#]+#]]&] (* Harvey P. Dale, Feb 19 2020 *)

A045855 Numbers whose square has initial digit '1'.

Original entry on oeis.org

1, 4, 10, 11, 12, 13, 14, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136

Offset: 1

Jeff Burch

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

Cf. A045784, A272671.

Select[Range[150],First[IntegerDigits[#^2]]==1&] (* Harvey P. Dale, Aug 28 2012 *)

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

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A273364 Numbers n such that the decimal number concat(9,n) is a square.

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A045855 Numbers whose square has initial digit '1'.

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

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

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