A058474 -id:A058474 - cp's OEIS frontend

A277959 Numbers k such that 2 is the largest decimal digit of k^2.

11, 101, 110, 149, 1001, 1010, 1011, 1100, 1101, 1490, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 14499, 14900, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 144990, 149000, 316261

Offset: 1

Colin Barker, Nov 06 2016

The terms > 1 of A058411 can be considered as primitive elements of this sequence, obtained by multiplying those by powers of 10 (cf. formula). These terms of A058411 have at least 2 nonzero digits, and therefore their square has at least one digit 2. - M. F. Hasler, Nov 15 2017

M. F. Hasler, Table of n, a(n) for n = 1..1380 (first 50 terms from Colin Barker)
OEIS Index to sequences related to squares.

Cf. A277946 (the squares); A277960, A277961, A295005, ..., A295009 (analog for largest digit 3, 4, 5, ..., 9).
Cf. A058411, A058412 and A058413, ..., A058474. (Similar but no trailing 0's allowed.)
Cf. A136808 and A136809, ..., A137147 for other digit combinations. (Numbers must satisfy the same restriction as their squares.)

Select[Range[4*10^5], And[#[[2]] > 0, Union@ Take[RotateLeft[#, 2], 7] == {0}] &@ DigitCount[#^2] &] (* Michael De Vlieger, Nov 16 2017 *)

L=List(); for(n=1, 10000, if(vecmax(digits(n^2))==2, listput(L, n))); Vec(L)

A277959(LIM=1e15, L=List(), N=1)={while(LIM>N=next_A058411(N),my(t=N); until(LIMM. F. Hasler, Nov 15 2017

Equals (A058411 \ {1})*A011557, where A011557 = { 10^k; k >= 0 }. - M. F. Hasler, Nov 16 2017

Edited by M. F. Hasler, Nov 16 2017

A058411 Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.

Original entry on oeis.org

1, 11, 101, 149, 1001, 1011, 1101, 10001, 10011, 11001, 14499, 100001, 100011, 100101, 101001, 110001, 316261, 1000001, 1000011, 1000101, 1010001, 1010011, 1100001, 1100101, 10000001, 10000011, 10000101, 10001001, 10001011, 10001101, 10010001, 10100001, 10100011, 10110001

Offset: 1

Patrick De Geest, Nov 15 2000

Sporadic solutions (not consisting only of digits 0 and 1): a(4) = 149, a(11) = 14499, a(17) = 316261, a(209) = 4604367505011, a(715) = 10959977245460011, a(1015) = 110000500908955011, a(1665) = 10099510939154979751, ... Three infinite subsequences are given by numbers of the form 10...01, 10...011 and 110...01, but there are many others. - M. F. Hasler, Nov 14 2017
From Zhao Hui Du, Mar 12 2024: (Start)
Most terms have a special pattern in that they have only digits 0 and 1 and could be written as Sum_{h=0..t} 10^x(h), where 2x(h) and x(h1)+x(h2) are distinct and x(0)=0 for the nonzero ending constraint. The number of n-digit terms in the sequence in the special pattern is A143823(n) - 2*A143823(n-1) + A143823(n-2) for n >= 2.
Terms with only digits 0 and 1 but not in the special pattern exist as well. If we define f(x) = 1 + x^768 + x^960 + x^1008 + x^1020 + x^1028 + x^1040 + x^1088 + x^1280 + x^2048, f(x)^2 is a function with all nonzero coefficients 1,2,10 (the only coefficient of x^2048 is 10 and the coefficient of x^2049 is 0). So f(10) is in the sequence but not in the special pattern. (End)

Zhao Hui Du, Table of n, a(n) for n = 1..4000 (first 1000 terms from Chai Wah Wu; 1001..1269 from M. F. Hasler)
Patrick De Geest, Index to related sequences.
Hisanori Mishima, Sporadic tridigital solutions.
OEIS Wiki, Index to sequences related to squares having only given digits.

Cf. A058412 (the squares); A058412, ..., A058474 (other 3-digit combinations).

Cf. A063009, A066139. - Zak Seidov, Jul 01 2013

[n: n in [1..2*10^8 by 2] | Set(Intseq(n^2)) subset [0,1,2]]; // Vincenzo Librandi, Feb 24 2016

R[1]:= {1,9};
for m from 2 to 10 do
  R[m]:= select(t -> max(convert(t^2 mod 10^m, base, 10)) <= 2, map(s -> seq(s + i*10^(m-1),i=0..9), R[m-1]))
od:
Res:= {seq(op(select(t -> t >= 10^(m-1) and max(convert(t^2,base,10)) <= 2, R[m])),m=1..10)}:
sort(convert(Res,list)); # Robert Israel, Feb 23 2016

Select[Range[10^6], And[Total@ Take[RotateRight@ DigitCount@ #, -7] == 0, Mod[#, 10] != 0] &[#^2] &] (* Michael De Vlieger, Nov 14 2017 *)

isok(n)={ n%10 && vecmax(digits(n^2)) < 3 } \\ Michel Marcus, Feb 24 2016, edited by M. F. Hasler, Nov 14 2017

A058411_list = [i for i in range(10**6) if i % 10 and max(str(i**2)) < '3'] # Chai Wah Wu, Feb 23 2016

a(n) = sqrt(A058412(n)). - Zak Seidov, Jul 01 2013

b-file corrected by Zhao Hui Du, Mar 07 2024

A137147 Numbers k such that k and k^2 use only the digits 5, 6, 7, 8 and 9.

Original entry on oeis.org

76, 87, 766, 887, 7666, 8887, 9786, 76587, 76666, 87576, 759576, 766666, 869866, 869867, 886886, 888587, 988866, 7666666, 8766867, 8885887, 76587576, 76666666, 76789686, 86998666, 87565786, 87685676, 88766867, 97759786, 97957576, 766666666, 875765766, 886885887, 887579686, 977699687

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

			989878759589576^2 = 979859958686597599779967859776.

Jonathan Wellons, Table of n, a(n) for n = 1..183
J. Wellons, Tables of Shared Digits [archived]
OEIS Index to sequences related to squares.

Cf. A136808, A136809, ..., A137146 for other digit combinations.
Cf. A000290 (the squares); A027675, A058411, ..., A058474 (3-digit combinations).
Cf. A277959, A277960, A277961, A295005, ..., A295009 (squares with largest digit = 2, 3, 4, 5, ..., 9).

A277948 Squares whose largest decimal digit is 4.

Original entry on oeis.org

4, 144, 324, 400, 441, 1024, 1444, 2304, 2401, 10404, 14400, 23104, 32041, 32400, 33124, 40000, 40401, 44100, 101124, 102400, 103041, 110224, 114244, 121104, 131044, 144400, 203401, 204304, 213444, 230400, 232324, 240100, 300304, 301401, 421201, 1004004

Offset: 1

Colin Barker, Nov 05 2016

A subsequence of A158082, in turn a subsequence of A000290.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000290 (the squares).
Cf. A277961 (square roots of these terms).
Cf. A277946, A277947, A295015, ..., A295019 (analog for largest digit = 2, 3, 5, ..., 9).
Cf. A058412, A058411, ..., A058474 and A136808, A136809, ..., A137147 for other restrictions on digits of squares.

[n^2: n in [1..1000000] | Maximum(Intseq(n^2)) eq 4]; // Vincenzo Librandi, Nov 06 2016

Select[Range[1100]^2,Max[IntegerDigits[#]]==4&] (* Harvey P. Dale, Jul 01 2017 *)

L=List(); for(n=1, 10000, if(vecmax(digits(n^2))==4, listput(L, n^2))); Vec(L)

a(n) = A277961(n)^2. - M. F. Hasler, Nov 12 2017

Intersection of A000290 and A277966. - M. F. Hasler, Nov 15 2017

A030175 When squared gives number composed of digits {1,2,3}.

Original entry on oeis.org

1, 11, 111, 36361, 363639, 461761, 3636361, 34815389, 362397739, 176412364139, 57637950363639, 3497458093147239, 56843832676142723489, 557963558954625926861

Offset: 1

Patrick De Geest

P. De Geest, Squares containing at most three distinct digits, Index entries for related sequences
Author?, Source(txt)
Hisanori Mishima, Sporadic tridigital solutions
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to squares.

Cf. A030174; A000290.
Cf. A006716 = A027675^2, A030176 = A030177^2, A058411^2 = A058412, ..., A058473^2 = A058474.
Cf. A136808, A136809, ..., A137147: n and n^2 have digits {...}.
Cf. A277959^2 = A277946 and A277960^2 = A277947: squares whose largest digit is 2 resp. 3.

Do[ If[ Union[ Join[{1, 2, 3}, IntegerDigits[n^2] ] ] == {1, 2, 3}, Print[n] ], {n, 0, 10^9}]

lista(nn) = for(n=1, nn, if(setminus(vecsort(digits(n^2), , 8), [1, 2, 3])==[], print1(n, ", "))) \\ Iain Fox, Nov 16 2017

a(n)^2 = A030174(n). - M. F. Hasler, Nov 16 2017

More terms from Patrick De Geest, Mar 01 2000
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 14 2005
Offset corrected by Iain Fox, Nov 16 2017

A006716 Squares with digits 1, 4, 9.

Original entry on oeis.org

1, 4, 9, 49, 144, 441, 1444, 11449, 44944, 991494144, 4914991449, 149991994944, 9141411499911441, 199499144494999441, 9914419419914449449, 444411911999914911441, 419994999149149944149149944191494441

Offset: 1

N. J. A. Sloane, revised Jul 10 2015

This is probably a finite sequence, but that is only a conjecture.

Since 1, 4 and 9 are squares, all terms are in A053059. - Rabii Younès, Mar 17 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 234.

Patrick De Geest, Squares containing at most three distinct digits, Index entries for related sequences
A. Ottens, The arithmetic-digits-squares-three.digits problem
Eric Weisstein's World of Mathematics, Square Number

Subsequence of A019544 and A053059.
Cf. A027675 (square roots), A061269.
For other digit groups {0,1,2} through {7,8,9}, see also: A058411, ..., A058472, A058473, A058474.

a(n) = A027675(n)^2. - M. F. Hasler, Nov 15 2017

a(13) corrected by Neven Juric (neven.juric(AT)apis-it.hr), May 14 2003

A058473 Numbers k such that k^2 contains only digits {0,1,9}, not ending with zero.

Original entry on oeis.org

1, 3, 43694278824566964251, 100990098979999970099500001

Offset: 1

Patrick De Geest, Nov 15 2000

P. De Geest, Index to related sequences
Hisanori Mishima, Sporadic tridigital solutions
Zhao Hui Du, Chinese BBS with updated data

Cf. A058474.

a(4) from Zhao Hui Du, Feb 28 2024

A136812 Numbers k such that k and k^2 use only the digits 0, 1, 2, 3 and 6.

Original entry on oeis.org

0, 1, 6, 10, 11, 60, 100, 101, 106, 110, 111, 361, 600, 601, 1000, 1001, 1006, 1010, 1011, 1060, 1100, 1101, 1106, 1110, 1631, 3606, 3610, 6000, 6001, 6010, 6011, 10000, 10001, 10006, 10010, 10011, 10060, 10100, 10101, 10106, 10110, 10111, 10301, 10306, 10600, 11000, 11001, 11006, 11010, 11060, 11100, 11101, 16310, 32111, 36060, 36100, 36361

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

			1031316261^2 = 1063613230203020121.

Jonathan Wellons, Table of n, a(n) for n = 1..1315
J. Wellons, Tables of Shared Digits [archived]
Index entries for sequences related to squares.

Cf. A136808, ..., A137147.

Cf. A006716 = A027675^2, A030174 = A030175^2, A030176 = A030177^2, A058411^2 = A058412, ..., A058473^2 = A058474.

A137146 Numbers k such that k and k^2 use only the digits 5, 6, 7 and 8.

Original entry on oeis.org

76, 766, 7666, 76666, 766666, 7666666, 76666666, 766666666, 7666666666, 76666666666, 766666666666, 7666666666666, 76666666666666, 766666666666666, 7666666666666666, 76666666666666666, 766666666666666666, 7666666666666666666, 76666666666666666666, 766666666666666666666

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

The first digit of each term is either 7 or 8 and the last digit is 6. - Chai Wah Wu, May 25 2021

			766666666666666^2 = 587777777777776755555555555556.

Jonathan Wellons, Tables of Shared Digits [archived].
OEIS Index to sequences related to squares.

Cf. A000290 (the squares); A136808, A136809, ..., A137147 for other digit combinations.
Cf. A058469 - A058472 and A058411, ..., A058474 for other digit combinations.
Cf. A277959, A277960, A277961, A295005, ..., A295009 (squares with largest digit = 2, 3, 4, 5, ..., 9).

from itertools import product
A137146_list = [n for n in (int(''.join(d)) for l in range(1,6) for d in product('5678',repeat=l)) if set(str(n**2)) <= set('5678')] # Chai Wah Wu, May 25 2021

a(15)-a(20) from Pontus von Brömssen, Apr 12 2024

cp's OEIS Frontend

A277959 Numbers k such that 2 is the largest decimal digit of k^2.

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A058411 Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.

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A137147 Numbers k such that k and k^2 use only the digits 5, 6, 7, 8 and 9.

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A277948 Squares whose largest decimal digit is 4.

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A030175 When squared gives number composed of digits {1,2,3}.

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A006716 Squares with digits 1, 4, 9.

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A058473 Numbers k such that k^2 contains only digits {0,1,9}, not ending with zero.

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A136812 Numbers k such that k and k^2 use only the digits 0, 1, 2, 3 and 6.

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