A277960 -id:A277960

A277959

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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

PROG

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

(PARI) A277959(LIM=1e15, L=List(), N=1)={while(LIM>N=next_A058411(N), my(t=N); until(LIM<t*=10, listput(L, t))); Set(L)} \\ M. F. Hasler, Nov 15 2017

CROSSREFS

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.)

KEYWORD

nonn,base

AUTHOR

Colin Barker, Nov 06 2016

EXTENSIONS

Edited by M. F. Hasler, Nov 16 2017

STATUS

approved

A277961

Numbers n such that 4 is the largest decimal digit of n^2.

+10
13

2, 12, 18, 20, 21, 32, 38, 48, 49, 102, 120, 152, 179, 180, 182, 200, 201, 210, 318, 320, 321, 332, 338, 348, 362, 380, 451, 452, 462, 480, 482, 490, 548, 549, 649, 1002, 1012, 1020, 1021, 1049, 1102, 1111, 1188, 1200, 1201, 1429, 1488, 1498, 1518, 1520

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The actual squares are listed in A277948. - M. F. Hasler, Nov 12 2017

Includes 2*10^n+10^m for all n <> m. - Robert Israel, Nov 13 2017

For any term of q digits, the first m digits don't exceed (2 * 10^m - 2) / 3 = 666..66 (m 6's) for 1 <= m <= q. - David A. Corneth, Nov 13 2017

A term a(n) is in the sequence if and only if a(n)*10^k is in the sequence, for all k >= 0. If a(n) = (x*10^k + y)*10^m with 2xy < 10^k, then (y*10^k+x)*10^m' is also in the sequence, for all m'. - M. F. Hasler, Nov 13 2017

LINKS

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

FORMULA

a(n) = sqrt(A277948(n)), where sqrt = A000196 or A000194 or A003059. - M. F. Hasler, Nov 12 2017

MAPLE

select(n -> max(convert(n^2, base, 10))=4, [$1..10000]); # Robert Israel, Nov 13 2017

PROG

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

CROSSREFS

Cf. A277948, A277959, A277960.

KEYWORD

nonn,base

AUTHOR

Colin Barker, Nov 06 2016

STATUS

approved

A137147

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Generated with DrScheme.

LINKS

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.

EXAMPLE

989878759589576^2 = 979859958686597599779967859776.

CROSSREFS

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).

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A030175

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..14.

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.

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

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.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest

EXTENSIONS

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

STATUS

approved

A277947

Squares whose largest decimal digit is 3.

+10
8

12321, 123201, 130321, 1232100, 1320201, 3101121, 12320100, 13032100, 102030201, 102232321, 103002201, 123210000, 123232201, 132020100, 310112100, 1232010000, 1303210000, 1322122321, 1332323001, 2103231321, 10022212321, 10130221201, 10203020100, 10203222121

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A subsequence of A000290.

LINKS

Colin Barker, Table of n, a(n) for n = 1..100

PROG

(PARI) L=List(); for(n=1, 10000, if(vecmax(digits(n^2))==3, listput(L, n^2))); Vec(L)

(Magma) [n^2: n in [1..1000000] | Maximum(Intseq(n^2)) eq 3]; // Vincenzo Librandi, Nov 06 2016

CROSSREFS

Cf. A000290.

Cf. A277946, A277948, A277960.

KEYWORD

nonn,easy,base

AUTHOR

Colin Barker, Nov 05 2016

STATUS

approved

A278937

Numbers k such that 3 is the largest decimal digit of k^3.

+10
8

11, 101, 110, 1001, 1010, 1100, 10001, 10010, 10100, 11000, 100001, 100010, 100100, 101000, 110000, 684917, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 6849170, 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A038444 is a subsequence. Are there an infinite number of terms not in A038444 that are not a multiple of 10? - Chai Wah Wu, Dec 02 2016

Conjecture: sequence is equal to A038444 plus terms of the form 684917*10^k for k >= 0. - Chai Wah Wu, Sep 02 2017

Conjecture is true up to 4.8*10^18. - Giovanni Resta, Sep 03 2017

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..130

FORMULA

a(n)^3 = A278936(n).

EXAMPLE

684917 is in the sequence because 684917^3 = 321302302131323213.

PROG

(PARI) select(n->vecmax(digits(n^3))==3, vector(1000000, n, n))

(Magma) [n: n in [1..2*10^7] | Max(Intseq(n^3)) eq 3]; // Vincenzo Librandi, Dec 03 2016

CROSSREFS

Cf. A000578 (the cubes: n^3), A038444, A277960 (analog for squares), A278936 (cubes of the terms: a(n)^3).

KEYWORD

nonn,base

AUTHOR

Colin Barker, Dec 02 2016

STATUS

approved

A295006

Numbers n such that the largest digit of n^2 is 6.

+10
4

4, 6, 8, 16, 19, 25, 34, 40, 46, 51, 56, 58, 60, 66, 68, 75, 79, 80, 81, 106, 108, 116, 119, 121, 125, 129, 142, 146, 156, 160, 162, 175, 190, 204, 206, 208, 215, 216, 225, 231, 238, 245, 246, 248, 249, 250, 251, 252, 254, 255, 256, 258, 325, 334, 340, 354, 355, 369, 375, 379

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = sqrt(A295016(n)), where sqrt = A000196 or A000194 or A003059.

EXAMPLE

19 is in this sequence because 19^2 = 361 has 6 as largest digit.

MATHEMATICA

Select[Range[400], Max[IntegerDigits[#^2]]==6&] (* Harvey P. Dale, Mar 30 2024 *)

PROG

(PARI) select( is_A295006(n)=n&&vecmax(digits(n^2))==6 , [0..999]) \\ The "n&&" avoids an error message for n=0.

CROSSREFS

Cf. A295016 (the corresponding squares), A277959, A277960, A277961, A295005 .. A295009 (analog for digits 2 through 9), A294996 (analog for cubes).

Cf. A000290 (the squares).

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Nov 12 2017

STATUS

approved

A137146

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..20.

Jonathan Wellons, Tables of Shared Digits [archived].

OEIS Index to sequences related to squares.

EXAMPLE

766666666666666^2 = 587777777777776755555555555556.

PROG

(Python)

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

CROSSREFS

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).

KEYWORD

base,nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved