A018827 -id:A018827 - cp's OEIS frontend

A018834 Numbers k such that the decimal expansion of k^2 contains k as a substring.

0, 1, 5, 6, 10, 25, 50, 60, 76, 100, 250, 376, 500, 600, 625, 760, 1000, 2500, 3760, 3792, 5000, 6000, 6250, 7600, 9376, 10000, 14651, 25000, 37600, 50000, 60000, 62500, 76000, 90625, 93760, 100000, 109376, 250000, 376000, 495475, 500000, 505025

Offset: 1

David W. Wilson

			25^2 = 625 which contains 25.
3792^2 = 14_3792_64, 14651^2 = 2_14651_801.

Giovanni Resta, Table of n, a(n) for n = 1..200 (first 126 terms from David W. Wilson)

Cf. A003226, A046831, A046851, A045953.
Cf. A000290. Supersequence of A029943.
Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9).
Cf. A029942 (cubes), A075904 (4th powers), A075905 (5th powers).

import Data.List (isInfixOf)
a018834 n = a018834_list !! (n-1)
a018834_list = filter (\x -> show x `isInfixOf` show (x^2)) [0..]
-- Reinhard Zumkeller, Jul 27 2011

Select[Range[510000], MemberQ[FromDigits /@ Partition[IntegerDigits[#^2], IntegerLength[#], 1], #] &] (* Jayanta Basu, Jun 29 2013 *)
Select[Range[0,510000],StringPosition[ToString[#^2],ToString[#]]!={}&] (* Ivan N. Ianakiev, Oct 02 2016 *)

from itertools import count, islice
def A018834_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:str(n) in str(n**2), count(max(startvalue,0)))
A018834_list = list(islice(A018834_gen(),20)) # Chai Wah Wu, Apr 04 2023

A018826 Numbers n such that n is a substring of its square when both are written in base 2.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 27, 32, 41, 54, 64, 82, 108, 128, 145, 164, 165, 256, 283, 290, 328, 487, 512, 545, 566, 580, 974, 1024, 1090, 1132, 1160, 1773, 1948, 2048, 2113, 2180, 2320, 2701, 3546, 3896, 4096, 4226, 4261, 4360, 4757, 5402, 7092, 7625, 8079, 8192

Offset: 1

David W. Wilson

Complement of A136492. - Reinhard Zumkeller, Jan 01 2008
A136510(a(n)) = 2 for n>0. - Reinhard Zumkeller, Jan 03 2008
From Robert Israel, Jul 11 2018: (Start)
Contains A000079.
If x satisfies x^2 == 8*x + 1 (mod 2^m) and 0 < x < 2^(m-3) then x is in the sequence. Note that x^2 == 8*x + 1 has 4 solutions mod 2^m for m >= 3. Terms obtained in this way include 27, 283, 1773, 9965, 55579, 206573, .... (End)

			27 in binary is 11011 and 27^2 = 729 in binary is 1011011001 which has substring 11011. - _Michael Somos_, Mar 16 2015

Giovanni Resta, Table of n, a(n) for n = 1..700 (first 200 terms from Robert Israel)

Cf. A000079, A076141, A136490, A136492, A136510.

Cf. A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9), A018834 (base 10).

filter:= proc(n) local S,S2;
    S:= convert(convert(n,binary),string);
    S2:= convert(convert(n^2,binary),string);
    StringTools:-Search(S,S2)<>0
end proc:
select(filter, [$0..10000]); # Robert Israel, Jul 11 2018

Select[Range[0, 8192], {} != SequencePosition @@ IntegerDigits[{#^2, #}, 2] &] (* Giovanni Resta, Aug 20 2018 *)
Select[Range[0,10000],SequenceCount[IntegerDigits[#^2,2],IntegerDigits[#,2]]>0&] (* Harvey P. Dale, May 03 2022 *)

issub(b, bs, k) = {for (i=1, #b, if (b[i] != bs[i+k-1], return (0));); return (1);}
a076141(n) = {if (n, b = binary(n), b = [0]); if (n, bs = binary(n^2), bs = [0]); sum(k=1, #bs - #b +1, issub(b, bs, k));}
lista(nn) = for (n=0, nn, if (a076141(n) == 1, print1(n, ", "))); \\ Michel Marcus, Mar 15 2015

def ok(n): return bin(n)[2:] in bin(n**2)[2:]
print([k for k in range(9999) if ok(k)]) # Michael S. Branicky, Apr 04 2024

A018828 Numbers n such that n is a substring of its square (both n and n squared in base 4) (written in base 10).

Original entry on oeis.org

0, 1, 4, 16, 41, 54, 64, 164, 165, 256, 290, 487, 545, 566, 1024, 1160, 1948, 2180, 3546, 4096, 4226, 4261, 7625, 8321, 8720, 9514, 13813, 13867, 15913, 16158, 16384, 16904, 33284, 46126, 54854, 55468, 63652, 64632, 65536, 66050, 67616, 113047, 130591

Offset: 1

David W. Wilson

Cf. A018826 (base 2), A018827 (base 3), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9), A018834 (base 10).

Select[Range[0,150000],SequenceCount[IntegerDigits[#^2,4],IntegerDigits[#,4]]>0&] (* Harvey P. Dale, Aug 13 2023 *)

Definition clarified by Harvey P. Dale, Aug 13 2023

A018829 Numbers n such that n is a substring of its square in base 5 (written in base 10).

Original entry on oeis.org

0, 1, 5, 14, 25, 38, 125, 349, 408, 543, 625, 3125, 4743, 5292, 10888, 12196, 13201, 15625, 25509, 26460, 36536, 43614, 55038, 57837, 78125, 136888, 207889, 219698, 234783, 390625, 445663, 1090347, 1098490, 1336564, 1437874, 1720752, 1915227

Offset: 1

David W. Wilson

Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9), A018834 (base 10).

Select[Range[0,2*10^6],SequenceCount[IntegerDigits[ #^2,5],IntegerDigits[ #,5]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 04 2019 *)

A018830 Numbers n such that n is a substring of its square in base 6 (written in base 10).

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 18, 24, 28, 36, 54, 81, 108, 136, 144, 168, 216, 324, 486, 648, 816, 864, 1008, 1216, 1296, 1944, 2916, 3888, 4896, 5184, 6048, 6458, 6561, 7296, 7776, 8451, 11664, 16768, 17496, 22779, 23328, 23985, 29376, 29889, 31104, 34299, 34549, 36288

Offset: 1

David W. Wilson

Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018831 (base 7), A018832 (base 8), A018833 (base 9), A018834 (base 10).

Select[Range[0, 10^5], StringContainsQ[IntegerString[#^2, 6], IntegerString[#, 6]] &] (* Paolo Xausa, Apr 05 2024 *)

A018831 Numbers n such that n is a substring of its square in base 7 (written in base 10).

Original entry on oeis.org

0, 1, 7, 30, 49, 285, 343, 911, 1900, 2208, 2401, 13962, 16807, 59763, 64098, 69128, 85880, 97734, 117649, 195032, 418341, 754422, 823543, 2162126, 2629229, 2790841, 3488518, 3842400, 4960401, 5764801, 7923812, 8559490, 15134882, 19765943, 31466333, 36415297

Offset: 1

David W. Wilson

Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018832 (base 8), A018833 (base 9), A018834 (base 10).

Select[Range[0, 10^7], StringContainsQ[IntegerString[#^2, 7], IntegerString[#, 7]] &] (* Paolo Xausa, Apr 05 2024 *)

a(35)-a(36) from Pontus von Brömssen, Apr 04 2024

A018832 Numbers n such that n is a substring of its square in base 8 (written in base 10).

Original entry on oeis.org

0, 1, 8, 27, 64, 283, 290, 512, 545, 1948, 2320, 4096, 4360, 8452, 13813, 16158, 16642, 23063, 26139, 27427, 27734, 32768, 33025, 67616, 87723, 110503, 129264, 130591, 133136, 184504, 206573, 226094, 262144, 264200, 526340, 701784, 1044728, 1050626

Offset: 1

David W. Wilson

Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018833 (base 9), A018834 (base 10).

Select[Range[0, 10^6], StringContainsQ[IntegerString[#^2, 8], IntegerString[#, 8]] &] (* Paolo Xausa, Apr 05 2024 *)

A018833 Numbers n such that n is a substring of its square in base 9 (written in base 10).

Original entry on oeis.org

0, 1, 9, 32, 43, 81, 287, 706, 729, 4170, 6561, 30433, 37530, 38669, 42783, 59049, 99543, 192211, 281782, 394981, 415578, 531441, 561785, 666112, 1723076, 2046242, 3039924, 3485488, 4782969, 8684182, 11512384, 12684634, 18346925, 19988197, 29728486, 31190295

Offset: 1

David W. Wilson

Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018834 (base 10).

Select[Range[0, 10^7], StringContainsQ[IntegerString[#^2, 9], IntegerString[#, 9]] &] (* Paolo Xausa, Apr 05 2024 *)

a(35)-a(36) from Pontus von Brömssen, Apr 04 2024

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A018834 Numbers k such that the decimal expansion of k^2 contains k as a substring.

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A018826 Numbers n such that n is a substring of its square when both are written in base 2.

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A018828 Numbers n such that n is a substring of its square (both n and n squared in base 4) (written in base 10).

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A018829 Numbers n such that n is a substring of its square in base 5 (written in base 10).

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A018830 Numbers n such that n is a substring of its square in base 6 (written in base 10).

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A018831 Numbers n such that n is a substring of its square in base 7 (written in base 10).

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A018832 Numbers n such that n is a substring of its square in base 8 (written in base 10).

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A018833 Numbers n such that n is a substring of its square in base 9 (written in base 10).

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