A269250 -id:A269250 - cp's OEIS frontend

A086008 Number of 0's in decimal expansion of n^2.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 1, 4, 2, 2, 2, 1

Offset: 0

Jason Earls, Jul 07 2003

			10^2 = 100, so a(10)=2 and 32^2 = 1024, so a(32)=1.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. 1's A086009, 2's A086010, 3's A086011, 4's A086012, 5's A086013, 6's A086014, 7's A086015, 8's A086016, 9's A086017.

Cf. A269250 for the n^3 analog.

DigitCount[(Range[0, 100])^2, 10, 0] (* G. C. Greubel, Dec 13 2016 *)

A086008(n)=!n+#select(d->!d,digits(n^2)) \\ M. F. Hasler, Feb 21 2016

A036527 Smallest cube containing exactly n 0's.

Original entry on oeis.org

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

A048365 a(n)^3 is smallest cube containing exactly n 0's.

Original entry on oeis.org

1, 0, 52, 10, 160, 520, 100, 1600, 5200, 1000, 16000, 52000, 10000, 160000, 520000, 100000, 1600000, 5200000, 1000000, 16000000, 52000000, 10000000, 160000000, 520000000, 100000000, 1600000000, 5200000000, 1000000000

Offset: 0

Patrick De Geest, Mar 15 1999

a(n) is the index of the first occurrence of n in A269250.-- Is there an index n = 3k+1 or n = 3k+2 from which on the pattern a(3k+1) = 16*10^k resp. a(3k+2) = 52*10^k is no longer true? - M. F. Hasler, Feb 20 2016

Eric Weisstein's World of Mathematics, Cubic Number

Cf. A036527, A269250.

nsmall = Table[Infinity, 12];
For[i = 0, i <= 52000, i++, n0 = Count[IntegerDigits[i^3], 0];
  If[nsmall[[n0]] > i, nsmall[[ n0]] = i]];
Join[{1}, nsmall]  (* Robert Price, Sep 26 2018 *)

a(3k) = 10^k; a(3k+1) <= 16*10^k (k>0), a(3k+2) <= 52*10^k. - M. F. Hasler, Feb 20 2016

a(19)-a(27) from Lars Blomberg, Jun 12 2011

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))
```

A269242 Number of times the digit 2 appears in the decimal expansion of n^3.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036529, i.e., A036529(n)^(1/3) = A048367(n) is the index of the first occurrence of n.

			0^3 = 0 has a(0) = 0 digits '2'.
1^3 = 1 has a(1) = 0 digits '2'.
2^3 = 8 has a(2) = 0 digits '2'.
3^3 = 27 has a(3) = 1 digits '2'.
4^3 = 64 has a(4) = 0 digits '2'.
5^3 = 125 has a(5) = 1 digit '2'.
28^3 = 21952 is the least cube which has a(28) = 2 digits '2'.

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

Cf. A036529 and A036527 - A036536; A048367 and A048365 - A048374.
Analog for the other digits 0, 1, ..., 9: A269250, A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086010 (digit 2), and A086008 - A086017 for digits 0 - 9.

[Multiplicity(Intseq(n^3),2): n in [0..100]]; // Marius A. Burtea, Jan 26 2020

seq(numboccur(2,convert(n^3,base,10)),n=0..100); # Robert Israel, Jan 26 2020

Table[DigitCount[n^3, 10, 2], {n, 0, 100}] (* Robert Price, Mar 21 2020 *)

A269242(n)=#select(t->t==2,digits(n^3))

A269243 Number of times the digit 3 appears in the decimal expansion of n^3.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036530, i.e., A036530(n)^(1/3) = A048368(n) is the index of the first occurrence of n.

			0^3 = 0, 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125 and 6^3 = 216 all have a(0) = a(1) = ... = a(6) = 0 digits '3'.
7^3 = 343 has a(7) = 2 digits '3'.

Cf. A036530 and A036527 - A036536; A048368 and A048365 - A048374.
Analog for the other digits 0, 1, ..., 9: A269250, A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086011 (digit 3), and A086008 - A086017 for digits 0 - 9.

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

PARI
```
A269243(n)=#select(t->t==3,digits(n^3))
```

A269244 Number of times the digit 4 appears in the decimal expansion of n^3.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036531, i.e., A036531(n)^(1/3) = A048369(n) is the index of the first occurrence of n.

			0^3 = 0, 1^3 = 1, 2^3 = 8 and 3^3 = 27 all have a(0) = a(1) = a(2) = a(3) = 0 digits '4'.
4^3 = 64 has a(4) = 1 digit '4'.
14^3 = 2744 has a(14) = 2 digits '4'.

Cf. A036531 and A036527 - A036536; A048369 and A048365 - A048374.
Analog for the other digits 0, 1, ..., 9: A269250, A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086012 (digit 4), and A086008 - A086017 for digits 0 - 9.

Table[DigitCount[n^3, 10, 4], {n, 0, 100}] (* Robert Price, Mar 21 2020 *)

PARI
```
A269244(n)=#select(t->t==4,digits(n^3))
```

A269245 Number of times the digit 5 appears in the decimal expansion of n^3.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036532, i.e., A036532(n)^(1/3) = A048370(n) is the index of the first occurrence of n.

			0^3 = 0, 1^3 = 1, 2^3 = 8 and 3^3 = 27 all have a(0) = a(1) = a(2) = a(3) = 0 digits '5'.
5^3 = 125 has a(5) = 1 digit '5'.

Cf. A036532 and A036527 - A036536; A048370 and A048365 - A048374.
Analog for the other digits 0, 1, ..., 9: A269250, A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086013 (digit 5), and A086008 - A086017 for digits 0 - 9.

Table[DigitCount[n^3, 10, 5], {n, 0, 100}] (* Robert Price, Mar 21 2020 *)

PARI
```
A269245(n)=#select(t->t==5,digits(n^3))
```

A269246 Number of times the digit 6 appears in the decimal expansion of n^3.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 3, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 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 A036533, i.e., A036533(n)^(1/3) = A048371(n) is the index of the first occurrence of n.

			0^3 = 0, 1^3 = 1, 2^3 = 8 and 3^3 = 27 all have a(0) = a(1) = a(2) = a(3) = 0 digits '6'.
4^3 = 64 has a(4) = 1 digit '6'.

Cf. A036533 and A036527 - A036536; A048371 and A048365 - A048374.
Analog for the other digits 0, 1, ..., 9: A269250, A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086014 (digit 6), and A086008 - A086017 for digits 0 - 9.

Table[DigitCount[n^3, 10, 6], {n, 0, 100}] (* Robert Price, Mar 21 2020 *)

PARI
```
A269246(n)=#select(t->t==6,digits(n^3))
```

A269247 Number of times the digit 7 appears in the decimal expansion of n^3.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Feb 20 2016

The cubes corresponding to the first occurrence of 1, 2, 3, ... are listed in A036534, i.e., A036534(n)^(1/3) = A048372(n) is the index of the first occurrence of n.

			0^3 = 0, 1^3 = 1, 2^3 = 8 and 4^3 = 64 all have a(0) = a(1) = a(2) = a(4) = 0 digits '7'.
3^3 = 27 has a(3) = 1 digit '7'.

Cf. A036534 and A036527 - A036536; A048372 and A048365 - A048374.
Analog for the other digits 0, 1, ..., 9: A269250, A269241, A269242, A269243, A269244, A269245, A269246, A269247, A269248, A269249.
Analog for squares: A086015 (digit 7), and A086008 - A086017 for digits 0 - 9.

Table[DigitCount[n^3, 10, 7], {n, 0, 100}] (* Robert Price, Mar 21 2020 *)

PARI
```
A269247(n)=#select(t->t==7,digits(n^3))
```

cp's OEIS Frontend

A086008 Number of 0's in decimal expansion of n^2.

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A036527 Smallest cube containing exactly n 0's.

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A048365 a(n)^3 is smallest cube containing exactly n 0's.

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

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A269242 Number of times the digit 2 appears in the decimal expansion of n^3.

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A269243 Number of times the digit 3 appears in the decimal expansion of n^3.

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A269244 Number of times the digit 4 appears in the decimal expansion of n^3.

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A269245 Number of times the digit 5 appears in the decimal expansion of n^3.

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