A104315 -id:A104315 - cp's OEIS frontend

A104316 A104315(n)^2.

11236, 11449, 11664, 11881, 41616, 42436, 42849, 43264, 43681, 92416, 93636, 94249, 94864, 95481, 163216, 164836, 165649, 166464, 167281, 364816, 367236, 368449, 369664, 495616, 498436, 499849, 646416, 649636, 651249, 652864, 654481

Offset: 1

Reinhard Zumkeller, Mar 01 2005

Subsequence of A052041, squares having in their decimal representation no zeros.

Cf. A104317.

A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

Original entry on oeis.org

0, 1, 2, 3, 10, 20, 30, 100, 200, 245, 247, 249, 251, 253, 283, 300, 448, 548, 949, 1000, 1249, 1253, 1416, 1747, 1749, 1751, 1753, 1755, 2000, 2245, 2247, 2249, 2251, 2253, 2429, 2450, 2451, 2470, 2490, 2498, 2510, 2530, 2647, 2830, 3000, 3747, 3751, 4480, 4899

Offset: 1

Bernard Schott, Oct 27 2019

The idea of this sequence comes from the 1st problem of the 28th British Mathematical Olympiad in 1992 (see the link).
This sequence is infinite because the family of integers {10^k, k >= 0} (A011557) belongs to this sequence.
The numbers m, m + 1, m + 2 where m = 49*10^k - 3, or m = 99*10^k - 3, k >= 3 are terms with all nonzero digits. - Marius A. Burtea, Dec 21 2020

			247^2 = 61009, hence 247 and 61009 both have 3 nonzero digits, 247 is a term.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 57 and 109 (1992)

Giovanni Resta, Table of n, a(n) for n = 1..10000.
British Mathematical Olympiad, 1992 - Problem 1.
Index to sequences related to Olympiads.

Subsequences: A011557, A093136, A093138.
Cf. A052040, A104315, A134844.
Cf. A328781, A328782, A328783.

nz:=func; [k:k in [0..5000] | nz(k) eq nz(k^2)]; // Marius A. Burtea, Dec 21 2020

q:= n->(f->f(n)=f(n^2))(t->nops(subs(0=[][], convert(t, base, 10)))):
select(q, [$0..5000])[];  # Alois P. Heinz, Oct 27 2019

Select[Range[0, 5000], Equal @@ Total /@ Sign@ IntegerDigits[{#, #^2}] &] (* Giovanni Resta, Feb 27 2020 *)

isok(k) = hammingweight(digits(k)) == hammingweight(digits(k^2)); \\ Michel Marcus, Dec 22 2020

More terms from Alois P. Heinz, Oct 27 2019

A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 54, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 96, 104, 105

Offset: 1

Bernard Schott, Oct 27 2019

Inspired by A328780.
This sequence is not a duplicate of A052040. The first 72 terms until 96 are exactly the same but a(73) = 104 belongs to this sequence because 104^2 = 10816, but 104 doesn't belong to A052040 because there is one zero digit in the decimal expansion of 104^2.
The nonnegative integers that do not belong to this sequence are divided into three sequences:
1) A104315 = A052040 \ {this sequence}: Numbers k such that k contains at least one zero, but k^2 contains no zero (e.g., 106 with 106^2 = 11236).
2) A134844 = Numbers k such that k contains no zero but k^2 contains at least one zero (e.g., 32 with 32^2 = 1024).
3) A328783 = Numbers k such that k and k^2 contain at least one zero but not the same number of zeros (e.g., 101 with 101^2 = 10201).
Another sequence is A328782 = {this sequence} \ A052040 which lists the positive integers that have the same positive number of zeros in their decimal expansions as in their squares. The first two examples > 0 are 104 with 104^2 = 10816 and 105 with 105^2 = 11025.

			12 and 144 = 12^2 have no digit zero in their decimal representation, so 12 is a term.
203 and 41209 = 203^2 both have one digit zero in their decimal representation, so 203 is also a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A052040, A104315, A134844.

Cf. A323780, A328782, A328783.

select(t -> numboccur(0, convert(t^2,base,10))=numboccur(0, convert(t,base,10)), [$0..200]); # Robert Israel, Oct 27 2019

Select[Range[0, 105], Equal @@ Total /@ (1 - Sign@ IntegerDigits[{#, #^2}]) &] (* Giovanni Resta, Feb 27 2020 *)

A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

Original entry on oeis.org

0, 104, 105, 203, 205, 302, 303, 305, 402, 403, 405, 504, 505, 506, 507, 508, 509, 601, 602, 603, 605, 609, 701, 702, 703, 705, 708, 709, 801, 802, 803, 805, 901, 902, 903, 905, 906, 1006, 1007, 1008, 1009, 1011, 1012, 1013, 1014, 1016, 1017, 1018, 1019, 1021

Offset: 1

Bernard Schott, Oct 28 2019

			703 and 494209 = 703^2 both have one zero digit in their decimal expansion.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Equals A328781 \ A052040.

Cf. A104315, A134844, A328780, A328783.

f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and x=y)(f(n), f(n^2)):
select(q, [$0..1030])[];  # Alois P. Heinz, Oct 28 2019

Select[Range[0, 1100], DigitCount[#, 10, 0] == DigitCount[#^2, 10, 0] > 0 &] (* Giovanni Resta, Feb 27 2020 *)

More terms from Alois P. Heinz, Oct 28 2019

A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 401, 410, 420, 430, 440, 450, 460, 470

Offset: 1

Bernard Schott, Oct 28 2019

This sequence is one of the three sequences whose numbers k and k^2 don't contain the same number of 0, the two others are A104315 and A134844.

			201 and 40401 = 201^2 have both at least one zero but not the same number of 0 in their decimal expansion, hence, 201 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A052040, A104315, A134844, A328780, A328781, A328782.

f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and y>0 and x<>y)(f(n), f(n^2)):
select(q, [$0..500])[];  # Alois P. Heinz, Oct 28 2019

Select[Range[0, 470], (x = DigitCount[#, 10, 0]) > 0 && (y = DigitCount[ #^2, 10, 0]) > 0 && x != y &] (* Giovanni Resta, Feb 27 2020 *)

More terms from Alois P. Heinz, Oct 28 2019

A104317 Number of n-digit squares with no zero digits, having roots containing at least one zero.

Original entry on oeis.org

0, 0, 0, 0, 14, 35, 186, 446, 2151, 5579, 22348, 58927, 216816, 583410

Offset: 1

Reinhard Zumkeller, Mar 01 2005

a(n) < A104264(n).

			a(5) = #{11236=106, 11449=107^2, 11664=108^2, 11881=109^2,
41616=204^2, 42436=206^2, 42849=207^2, 43264=208^2, 43681=209^2,
92416=304^2, 93636=306^2, 94249=307^2, 94864=308^2, 95481=309^2} = 14.

Cf. A104316, A104315, A104264.

A234966 Least number k with at least one zero such that k^n contains no zero, or 0 if no such number exists.

Original entry on oeis.org

0, 106, 104, 104, 105, 102, 102, 408, 104, 107, 203, 109, 103, 103, 1056, 3703, 4604, 207, 606, 11018, 3069, 20064

Offset: 1

Derek Orr, Jan 02 2014

a(n) > 5*10^8 or 0 for n = 23 and for 25 < n < 75.
It is known that a(24) = 12801714 and a(25) = 402.
a(n) > 5*10^9 or 0 for n = 23 and for 25 < n <= 200. - Chai Wah Wu, Apr 25 2019

			a(5) = 105 because 105 is the smallest number with a 0 where 105^5 does not have a 0 (105^5 = 12762815625).

Cf. A104315.

def f(x):
    for n in range(10**7):
        if "0" in str(n):
            if "0" not in str(n**x):
                return n
for x in range(1, 75):
    if f(x) is None:
        print(0)
    else:
        print(f(x))

cp's OEIS Frontend

A104316 A104315(n)^2.

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A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

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A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

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A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

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A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

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A104317 Number of n-digit squares with no zero digits, having roots containing at least one zero.

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A234966 Least number k with at least one zero such that k^n contains no zero, or 0 if no such number exists.

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