A295009 -id:A295009 - 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

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).

A295005 Numbers n such that the largest digit of n^2 is 5.

Original entry on oeis.org

5, 15, 35, 39, 45, 50, 55, 65, 71, 105, 112, 115, 145, 150, 155, 185, 188, 205, 211, 229, 235, 335, 350, 365, 368, 388, 389, 390, 450, 461, 485, 495, 500, 501, 502, 505, 550, 579, 585, 595, 635, 650, 652, 665, 671, 710, 711, 715, 718, 729, 735, 745, 1005, 1015, 1050

Offset: 1

M. F. Hasler, Nov 12 2017

			39 is in this sequence because 39^2 = 1521 has 5 as largest digit.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A295015 (the corresponding squares), A277959 .. A277961 (same for digit 2 .. 4), A295006 .. A295009 (same for digit 6 .. 9).

Cf. A000290 (the squares).

Select[Sqrt[ #]&/@(FromDigits/@Select[Tuples[ Range[ 0,5],7],Max[#] == 5&]),IntegerQ] (* Harvey P. Dale, Sep 23 2021 *)

select( is_A295005(n)=n&&vecmax(digits(n^2))==5 , [0..999]) \\ The "n&&" avoids an error message for n=0.

def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    if max(str(k*k)) == "5": alst.append(k)
  return alst
print(aupto(1050)) # Michael S. Branicky, May 15 2021

a(n) = sqrt(A295015(n)), where sqrt = A000196 or A000194 or A003059.

A295019 Squares whose largest digit is 9.

Original entry on oeis.org

9, 49, 169, 196, 289, 529, 729, 900, 961, 1089, 1296, 1369, 1849, 1936, 2209, 2809, 2916, 3249, 3969, 4096, 4489, 4900, 5329, 5929, 6889, 7396, 7569, 7921, 8649, 9025, 9216, 9409, 9604, 9801, 10609, 11449, 12769, 12996, 13689, 13924, 15129, 16129, 16900, 17689, 17956, 18496, 18769

Offset: 1

M. F. Hasler, Nov 12 2017

Cf. A295009 (square roots of the terms), A277946 - A277948 (same for digit 2..4), A295015 - A295018 (same for digit 5..8).

Cf. A000290 (the squares).

Select[Range[150]^2,Max[IntegerDigits[#]]==9&] (* Harvey P. Dale, Oct 27 2019 *)

is_A295019(n)=issquare(n)&&n&&vecmax(digits(n))==9 \\ "&&n" avoids an error message for n=0.

a(n) = A295009(n)^2.

A295006 Numbers n such that the largest digit of n^2 is 6.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Nov 12 2017

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

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

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

Cf. A000290 (the squares).

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

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

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

A295008 Numbers whose square has largest digit 8.

Original entry on oeis.org

9, 22, 28, 29, 41, 59, 62, 72, 78, 90, 91, 92, 94, 104, 109, 122, 126, 128, 135, 151, 159, 168, 169, 178, 184, 191, 192, 195, 196, 202, 209, 220, 221, 232, 241, 242, 259, 261, 262, 268, 278, 279, 280, 284, 285, 289, 290, 291, 292, 294, 295, 296, 298, 322, 328, 329, 341, 344, 349

Offset: 1

M. F. Hasler, Nov 12 2017

Includes a*10^n+b for n >= 2 and [a,b] in {[4, 1], [9, 1], [2, 2], [9, 2], [1, 4], [6, 4], [9, 4], [8, 5], [4, 6], [9, 6], [5, 8], [8, 8], [9, 8], [1, 9], [2, 9], [4, 9], [6, 9], [8, 9], [9, 9]}. - Robert Israel, Nov 13 2017

			28 is in this sequence because 28^2 = 784 has 8 as largest digit.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A295018 (the corresponding squares), A277959 .. A277961 (same for digit 2 .. 4), A295005 .. A295009 (same for digit 5 .. 9).

Cf. A000290 (the squares).

select(t -> max(convert(t^2,base,10))=8, [$1..1000]); # Robert Israel, Nov 13 2017

Select[Range[400],Max[IntegerDigits[#^2]]==8&] (* Harvey P. Dale, Jun 02 2019 *)

select( is_A295008(n)=n&&vecmax(digits(n^2))==8 , [0..999]) \\ The "n&&" avoids an error message for n=0.

def ok(n): return max(int(d) for d in str(n*n)) == 8
print(list(filter(ok, range(350)))) # Michael S. Branicky, Sep 22 2021

a(n) = sqrt(A295018(n)), where sqrt = A000196 or A000194 or A003059.

A322490 Numbers k such that k^k ends with 7.

Original entry on oeis.org

3, 17, 23, 37, 43, 57, 63, 77, 83, 97, 103, 117, 123, 137, 143, 157, 163, 177, 183, 197, 203, 217, 223, 237, 243, 257, 263, 277, 283, 297, 303, 317, 323, 337, 343, 357, 363, 377, 383, 397, 403, 417, 423, 437, 443, 457, 463, 477, 483, 497, 503, 517, 523, 537, 543, 557, 563

Offset: 1

Bruno Berselli, Dec 12 2018

Equivalently, numbers k such that k and (7^h)^k end with the same digit, where h == 1 (mod 4).
Also, numbers k such that k and (3^h)^k end with the same digit, where h == 3 (mod 4).
Numbers congruent to {3, 17} mod 20. - Amiram Eldar, Feb 27 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A056849.
Subsequence of A063226, A295009.
Similar sequences are listed in A322489.

GAP
```
List([1..70], n -> 10*n+2*(-1)^n-5);
```

[10*n+2*(-1)^n-5 for n in 1:70] |> println

Magma
```
[10*n+2*(-1)^n-5: n in [1..70]];
```

select(n->n^n mod 10=7,[$1..563]); # Paolo P. Lava, Dec 18 2018

Table[10 n + 2 (-1)^n - 5, {n, 1, 60}]
LinearRecurrence[{1,1,-1},{3,17,23},80] (* Harvey P. Dale, Sep 15 2019 *)

Maxima
```
makelist(10*n+2*(-1)^n-5, n, 1, 70);
```

apply(A322490(n)=10*n+2*(-1)^n-5, [1..70])

Vec(x*(3 + 14*x + 3*x^2) / ((1 + x)*(1 - x)^2) + O(x^55)) \\ Colin Barker, Dec 13 2018

[10*n+2*(-1)**n-5 for n in range(1, 70)]

Sage
```
[10*n+2*(-1)^n-5 for n in (1..70)]
```

O.g.f.: x*(3 + 14*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: 3 + 2*exp(-x) + 5*(2*x - 1)*exp(x).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 10*n + 2*(-1)^n - 5. Therefore:
a(n) = 10*n - 7 for odd n;
a(n) = 10*n - 3 for even n.
a(n+2*k) = a(n) + 20*k.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(7*Pi/20)*Pi/20. - Amiram Eldar, Feb 27 2023

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

A295007 Numbers n such that the largest digit of n^2 is 7.

Original entry on oeis.org

24, 26, 42, 52, 61, 69, 74, 76, 82, 84, 85, 88, 124, 131, 132, 144, 154, 165, 166, 174, 181, 189, 194, 218, 224, 226, 234, 239, 240, 260, 265, 266, 269, 271, 274, 275, 276, 319, 326, 356, 371, 376, 384, 415, 416, 418, 419, 420, 421, 448, 455, 466, 474, 476, 520, 521, 524, 525, 526, 552

Offset: 1

M. F. Hasler, Nov 12 2017

			24 is in this sequence because 24^2 = 576 has 7 as largest digit.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A295017 (the corresponding squares), A277959 .. A277961 (same for digit 2 .. 4), A295005 .. A295009 (same for digit 5 .. 9).

Cf. A000290 (the squares).

filter:= proc(n) max(convert(n^2,base,10))=7 end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 19 2019

select( is_A295007(n)=n&&vecmax(digits(n^2))==7 , [0..999]) \\ The "n&&" avoids an error message for n=0.

a(n) = sqrt(A295017(n)), where sqrt = A000196 or A000194 or A003059.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A295005 Numbers n such that the largest digit of n^2 is 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A295019 Squares whose largest digit is 9.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A295006 Numbers n such that the largest digit of n^2 is 6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A295008 Numbers whose square has largest digit 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A322490 Numbers k such that k^k ends with 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Julia

Magma

Maple