A036528 -id:A036528 - cp's OEIS frontend

A036527 Smallest cube containing exactly n 0's.

1, 0, 140608, 1000, 4096000, 140608000, 1000000, 4096000000, 140608000000, 1000000000, 4096000000000, 140608000000000, 1000000000000, 4096000000000000, 140608000000000000, 1000000000000000, 4096000000000000000, 140608000000000000000, 1000000000000000000, 4096000000000000000000, 140608000000000000000000, 1000000000000000000000

Offset: 0

David W. Wilson

a(n)^(1/3) = A048365(n) is the index of the first occurrence of n in A269250. -- For n = 3k, obviously a(n) = 10^n. The first terms for indices n = 3k+1 and n = 3k+2 equals 4096*10^3k resp. 140608*10^3k. Is there an index from where on this is no longer true? - M. F. Hasler, Feb 20 2016

Cf. A269250, A086008, A048365.
Cf. A036528 - A036536 for other digits 1 - 9.
Analog for squares: A036507 = A048345^2.

nsmall = Table[Infinity, 20];
For[i = 0, i <= 10^6, i++, n0 = Count[IntegerDigits[i^3], 0];
  If[nsmall[[n0 + 1]] > i^3, nsmall[[n0 + 1]] = i^3]];
Cases[nsmall, ?NumberQ] (* _Robert Price, Mar 20 2020 *)

a(n) = A048365(n)^3; a(3n) = 10^(3n); a(3n+1) <= 4096*10^(3n) = (16*10^n)^3 for n>0; a(3n+2) <= 140608*10^(3n) = (52*10^n)^3, with equality for all known terms. - M. F. Hasler, Feb 20 2016

Extended to a(0) = 1 and three lines of data completed by M. F. Hasler, Feb 20 2016

A269250 Number of times the digit 0 appears in the decimal expansion of n^3.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 2, 0, 0, 0, 2, 0, 1, 3, 0, 0, 1, 1, 0, 0, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 1, 0, 0, 0, 0, 1, 6

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036527, i.e., A048365(n) = A036527(n)^(1/3) is the index of the first occurrence of n.

			1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, ..., 9^3 = 729 all have a(1) = a(2) = ... = a(9) = 0 digits '0'.
0^3 = 0 has a(0) = 1 digit '0'.
10^3 = 1000 has a(10) = 3 digits '0'.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A048365 = A036527^(1/3); A036528 - A036536 and A048366 - A048374.
Analog for the other digits 1, ..., 9: A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086008 (digit 0), and A086009 - A086017 for digits 1 - 9.

seq(numboccur(0, convert(n^3,base,10)), n=0..100); # Robert Israel, Feb 21 2016

Table[DigitCount[n^3, 10, 0], {n, 0, 100}] (* Robert Price, Mar 21 2020 *)

A269250(n)=!n+#select(t->!t,digits(n^3))

A269241 Number of times the digit 1 appears in the decimal expansion of n^3.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 3, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036528, i.e., A036528(n)^(1/3) = A048366(n) is the index of the first occurrence of n.

			0^3 = 0 has a(0) = 0 digits '1'.
1^3 = 1 has a(1) = 1 digit '1'.
2^3 = 8 has a(2) = 0 digits '1'.
3^3 = 27 has a(3) = 0 digits '1'.
4^3 = 64 has a(4) = 0 digits '1'.
5^3 = 125 has a(5) = 1 digit '1'.
11^3 = 1331 is the smallest cube to have a(11) = 2 digits '1'.

Cf. A036528 and A036527 - A036536; A048366 and A048365 - A048374.
Analog for the other digits 0, 2, ..., 9: A269250, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086009 (digit 1), and A086008 - A086017 for digits 0 - 9.

Table[DigitCount[n^3, 10, 1], {n, 0, 99}] (* Alonso del Arte, Feb 20 2016 *)

PARI
```
A269241(n)=#select(t->t==1,digits(n^3))
```

A048366 a(n)^3 is smallest cube containing exactly n 1's.

Original entry on oeis.org

1, 11, 58, 106, 671, 1041, 10058, 22598, 145981, 480765, 2359231, 10297461, 4836178, 100395471, 465933117, 481182258, 4810215701, 16886336471, 49303833471, 103791158471, 223818432208, 4643311948655, 4809689791471

Offset: 1

Patrick De Geest, Mar 15 1999

Eric Weisstein's World of Mathematics, Cubic Number

Cf. A036528, A048365, A048367, A048368, A048369, A048370, A048371, A048372, A048373, A048374.

a[n_] := Module[{i}, i = 1; While[DigitCount[i^3][[1]] != n, i++;]; i]; (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 23 2006 *)

a(14) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 23 2006
a(15)-a(20) from Lars Blomberg, Jun 12 2011
a(21)-a(23) from Giovanni Resta, Jun 29 2018

A036529 Smallest cube containing exactly n 2's.

Original entry on oeis.org

0, 27, 21952, 2628072, 202262003, 229220928, 22521332224, 21148222722264, 25942222239227, 2272271222935232, 2262268226562252992, 4223937222222326272, 22225347273222227224, 122245292222422449622424, 2732072222242422541222208, 22422524292920620222272827

Offset: 0

David W. Wilson

a(n)^(1/3) = A048367(n) is the index of the first occurrence of n in sequence A269242. - M. F. Hasler, Feb 21 2016

Giovanni Resta, Table of n, a(n) for n = 0..23

Cf. A048367, A036527, A036528, A036530, A036531, A036532, A036533, A036534, A036535, A036536.

nsmall = Table[Infinity, 20];
For[i = 0, i <= 10^6, i++, n0 = Count[IntegerDigits[i^3], 2];
  If[nsmall[[n0 + 1]] > i^3, nsmall[[n0 + 1]] = i^3]];
Cases[nsmall, ?NumberQ] (* _Robert Price, Mar 20 2020 *)

a(n) = A048367(n)^3. - M. F. Hasler, Feb 21 2016

a(12)-a(15) from Giovanni Resta, Jun 29 2018

A036530 Smallest cube containing exactly n 3's.

Original entry on oeis.org

0, 4913, 343, 5735339, 108531333, 353393243, 333533450375, 3707322336333, 86333306433373, 153712433903333733, 333343033492332032, 8343304933332333329, 433134325352337333353, 33435346723033333302433, 243513939323833732333333, 33393753313938361233336383

Offset: 0

David W. Wilson

a(n)^(1/3) = A048368(n) is the index of the first occurrence of n in sequence A269243. - M. F. Hasler, Feb 21 2016

Giovanni Resta, Table of n, a(n) for n = 0..23

Cf. A048368, A036527, A036528, A036529, A036531, A036532, A036533, A036534, A036535, A036536.

Table[SelectFirst[Range[2100000]^3,DigitCount[#,10,3]==n&],{n,11}] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Oct 24 2015 *)

a(n) = A048368(n)^3. - M. F. Hasler, Feb 21 2016

a(0) = 0 prefixed by M. F. Hasler, Feb 21 2016

a(12)-a(15) from Giovanni Resta, Jun 29 2018

A036531 Smallest cube containing exactly n 4's.

Original entry on oeis.org

0, 64, 2744, 1481544, 4410944, 444194947, 44474744007, 4970444443496, 2440744441344, 4408846444574424, 434424163454644544, 40045354844444494784, 304443494462464444459, 24144094248434404444864, 45444444436448021414449, 442063442345444443482444864

Offset: 0

David W. Wilson

a(n)^(1/3) = A048369(n) is the index of the first occurrence of n in sequence A269244. - M. F. Hasler, Feb 21 2016

Giovanni Resta, Table of n, a(n) for n = 0..22

Cf. A048369, A036527, A036528, A036529, A036530, A036532, A036533, A036534, A036535, A036536.

nsmall = Table[Infinity, 20];
For[i = 0, i <= 10^6, i++, n0 = Count[IntegerDigits[i^3], 4];
  If[nsmall[[n0 + 1]] > i^3, nsmall[[n0 + 1]] = i^3]];
Cases[nsmall, ?NumberQ] (* _Robert Price, Mar 21 2020 *)

a(n) = A048369(n)^3. - M. F. Hasler, Feb 21 2016

Extended with a(0) = 0 by M. F. Hasler, Feb 21 2016

a(11)-a(15) from Giovanni Resta, Jun 29 2018

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A036527 Smallest cube containing exactly n 0's.

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A269250 Number of times the digit 0 appears in the decimal expansion of n^3.

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A269241 Number of times the digit 1 appears in the decimal expansion of n^3.

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A048366 a(n)^3 is smallest cube containing exactly n 1's.

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A036529 Smallest cube containing exactly n 2's.

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A036530 Smallest cube containing exactly n 3's.

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A036531 Smallest cube containing exactly n 4's.

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