A064532 -id:A064532 - cp's OEIS frontend

A359245 The smallest square with exactly n circular loops (or holes) in its decimal expansion (A064532).

1, 0, 81, 289, 1089, 8836, 6889, 80089, 688900, 1868689, 8508889, 29888089, 288898009, 983888689, 3808988089, 8680089889, 86908808809, 488088068689, 878686888689, 2888986888804, 48890888808804, 108506888888896, 88869893888889, 880881089888881, 788088668888889

Offset: 0

Bernard Schott, Dec 22 2022

The digit 8 has two loops, the digits 0, 6 and 9 have one loop, and other digits (including 4) have no hole.

Least square k such that A064532(k) = n.

			a(3) = 289 because 8 has two loops and 9 has one loop for a total of 3, and 289 is the smallest such square.

Cf. A064532.

Similar: A331898 (primes), A337842 (palindromes).

a(n) = { for (k=0, oo, my (d=digits(k^2)); if (n==(k==0)+sum(i=1, #d, [1, 0, 0, 0, 0, 0, 1, 0, 2, 1][1+d[i]]), return (k^2))) } \\ Rémy Sigrist, Dec 22 2022

More terms from Rémy Sigrist, Dec 22 2022

A064692 Total number of holes in decimal expansion of the number n, assuming 4 has 1 hole.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 3, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 1, 2

Offset: 0

Matthew Conroy, Oct 11 2001

			We assume decimal digits 1,2,3,5,7 have no hole; 0,4,6,9 have one hole each; 8 has two holes. So a(148)=3.

Indranil Ghosh, Table of n, a(n) for n = 0..50000

Cf. A001742, A064532, A064693, A206439, A064692, A208568.

Table[DigitCount[x].{0, 0, 0, 1, 0, 1, 0, 2, 1, 1}, {x, 0, 104}] (* Zak Seidov, Jul 25 2015 *)

def A064692(n):
    x=str(n)
    return x.count("0")+x.count("4")+x.count("6")+x.count("8")*2+x.count("9") # Indranil Ghosh, Feb 02 2017

a(A001742(n)) = 0. - Michel Marcus, Jul 25 2015

A064530 Number of holes in n-th capital letter of English alphabet.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 07 2001

			A has one hole, B has two, etc.

Index entries for sequences related to holes in letters

Equals A064529 - 1. Cf. A064531, A064532, A208568.

A064531 Number of connected components remaining when decimal expansion of the number n is cut from a piece of paper.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 4, 3, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 3, 3, 3, 3, 4, 3, 5, 4, 3, 2, 2, 2, 2, 2, 3, 2, 4, 3, 3, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane, Oct 07 2001

Assumes that 4 is represented without a hole.

Cf. A064529, A064530. Equals A064532 + 1.

a[n_ /; 0 <= n <= 9] := a[n] = {2, 1, 1, 1, 1, 1, 2, 1, 3, 2}[[n + 1]]; a[n_] := Total[a /@ (id = IntegerDigits[n])] - Length[id] + 1 ; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 22 2013 *)

More terms from Matthew Conroy, Oct 09 2001

A064529 Number of connected components remaining when n-th letter of English alphabet is cut from a piece of paper.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 07 2001

Index entries for sequences related to holes in letters

Equals A064530 + 1. Cf. A064531, A064532.

A331898 The smallest prime number with exactly n circular loops in its decimal representation.

Original entry on oeis.org

2, 19, 83, 89, 809, 1889, 8089, 48889, 88883, 828889, 688889, 3888889, 8868889, 28888889, 88888883, 288888889, 808888889, 6886888889, 8688888889, 48888888889, 188688888889, 288888888889, 888088888889, 1888888888889, 8888988888889, 58888888888889, 188880888888889

Offset: 0

Sara Mutter, Jan 31 2020

Least prime p such that A064532(p) = n.

The digit 8 has two loops and the digits 0, 6 and 9 have one loop.

			a(3) = 89 because 8 has two loops and 9 has one loop for a total of 3.

Giovanni Resta, Table of n, a(n) for n = 0..100
Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 19

Cf. A000040, A064532, A327820.

Block[{s = Range[0, 15]}, Sort[#][[All, -1]] &@ Reap[Do[If[! FreeQ[s, #2], Sow[{#2, #1}]; s = DeleteCases[s, #2]] & @@ {#, Total[{0, 0, 0, 0, 0, 1, 0, 2, 1, 1} DigitCount[#]]} &@ Prime@ i, {i, 3*10^5}]][[-1, -1]]] (* Michael De Vlieger, Feb 08 2020 *)
s[0]={1,2,3,4,5,7}; s[1]={0,6,9}; s[2]={8}; m[{sn_, t_}] := Union[Sort /@ Tuples[ s[sn], {t}]]; f[nd_, nh_] := Block[{v, pa = Tally /@ IntegerPartitions[ nh, {nd}, {0,1,2}], bst = Infinity}, Do[v = Flatten /@ Tuples[m /@ p]; Do[z = Select[ FromDigits /@ Select[ Permutations@ e, First[#] > 0 && OddQ@ Last@ # &], PrimeQ]; bst = Min[bst, {z}], {e, v}], {p, pa}]; bst]; a[0]=2; a[n_]:= Block[{nd = Ceiling[(n + 1)/2], b}, While[! IntegerQ@(b = f[nd, n]), nd++]; b]; a /@ Range[0, 30] (* Giovanni Resta, Feb 09 2020 *)

\\ here b(n) is A064532.
b(n)={vecsum([if(d==8,2, d==0||d==6||d==9) | d<-digits(n)])}
a(n)={forprime(p=1, oo, if(b(p)==n, return(p)))} \\ Andrew Howroyd, Jan 31 2020

a(13)-a(16) from Andrew Howroyd, Jan 31 2020
a(17)-a(19) from Jinyuan Wang, Feb 08 2020
a(20)-a(26) from Giovanni Resta, Feb 09 2020

A337842 The smallest palindrome with exactly n circular loops (or holes) in its decimal representation.

Original entry on oeis.org

1, 0, 8, 606, 88, 808, 888, 68086, 8888, 88088, 88888, 6880886, 888888, 8880888, 8888888, 688808886, 88888888, 888808888, 888888888, 68888088886, 8888888888, 88888088888, 88888888888, 6888880888886, 888888888888, 8888880888888, 8888888888888, 688888808888886

Offset: 0

Bernard Schott, Sep 25 2020

The decimal digits 1, 2, 3, 5, 7 have no hole, and 4 is represented without a hole; otherwise, 0, 6, 9 have one hole each and 8 has two holes.
Least palindrome q such that A064532(q) = n.
Except for a(0) = 1, each term has only digits 0, 6 or 8 in its decimal expansion.

			a(3) = 606 because 6 and 0 have each one circular loop for a total of 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to holes in digits
Index entries for linear recurrences with constant coefficients, signature (0,0,0,111,0,0,0,-1110,0,0,0,1000).

Cf. A000533, A002113, A002282, A064532, A332180.

Cf. A331898 (similar for primes).

a[0] = 1; a[n_] := Switch[Mod[n, 4], ?EvenQ, 8*(10^(n/2) - 1)/9, 1, 8*(10^((n + 1)/2) - 9*10^((n - 1)/4) - 1)/9, 3, 8*(10^((n + 3)/2) - 9*10^((n + 1)/4) - 1)/9 - 2*(10^((n + 1)/2) + 1)]; Array[a, 28, 0] (* _Amiram Eldar, Sep 26 2020 *)
LinearRecurrence[{0,0,0,111,0,0,0,-1110,0,0,0,1000},{1,0,8,606,88,808,888,68086,8888,88088,88888,6880886,888888},30] (* Harvey P. Dale, Mar 13 2022 *)

a(2m) = A002282(m) for m >= 1.
a(4m+1) = A332180(m) for m >= 1.
a(4m+3) = 6 * A000533(2m+2) + 10 * A332180(m) for m >= 0.

More terms from Amiram Eldar, Sep 25 2020

A358439 Number of even digits necessary to write all positive n-digit integers.

Original entry on oeis.org

4, 85, 1300, 17500, 220000, 2650000, 31000000, 355000000, 4000000000, 44500000000, 490000000000, 5350000000000, 58000000000000, 625000000000000, 6700000000000000, 71500000000000000, 760000000000000000, 8050000000000000000, 85000000000000000000, 895000000000000000000

Offset: 1

Bernard Schott, Nov 16 2022

If nonnegative n-digit integers were considered, then a(1) would be 5.
Also, a(n) is the total number of holes in all positive n-digit integers, assuming 4 has no hole. Digits 0, 6 and 9 have 1 hole, digit 8 has 2 holes, and other digits have no holes or (circular) loops (as in A064532).
Proof of the first formula: For n>=2, to write all positive n-digit integers, digits 6, 8, 9 occur A081045(n-1) = (9n+1)*10^(n-2) times each, and digit 0 occurs A212704(n-1) = 9*(n-1)*10^(n-2) times; so a(n) = 4*A081045(n-1) + A212704(n-1).
For a(1), if 0 were included then there would be 5 holes in the 1-digit numbers 0..9.

			To write the integers from 10 up to 99, each of the digits 2, 4, 6 and 8 must be used 19 times, and digit 0 must be used 9 times hence a(2) = 4*19 + 9 = 85.

Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A064532, A081045, A212704.

Cf. A113119, A196563, A358854, A359271.

Maple
```
seq((5*(9*n-1))*10^(n-2), n = 1 .. 30);
```

a[n_] := 5*(9*n - 1)*10^(n - 2); Array[a, 22] (* Amiram Eldar, Nov 16 2022 *)

a(n) = 5*(9n-1)*10^(n-2).
Formulas coming from the name with even digits:
a(n) = A358854(10^n-1) - A358854(10^(n-1)-1).
a(n) = A113119(n) - A359271(n) for n >= 2.
Formula coming from the comment with holes:
a(n) = Sum_{k=10^(n-1)..10^n-1} A064532(k).

cp's OEIS Frontend

A359245 The smallest square with exactly n circular loops (or holes) in its decimal expansion (A064532).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A064692 Total number of holes in decimal expansion of the number n, assuming 4 has 1 hole.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A064530 Number of holes in n-th capital letter of English alphabet.

Views

Author

Keywords

Examples

Links

Crossrefs

A064531 Number of connected components remaining when decimal expansion of the number n is cut from a piece of paper.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A064529 Number of connected components remaining when n-th letter of English alphabet is cut from a piece of paper.

Views

Author

Keywords

Links

Crossrefs

A331898 The smallest prime number with exactly n circular loops in its decimal representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A337842 The smallest palindrome with exactly n circular loops (or holes) in its decimal representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A358439 Number of even digits necessary to write all positive n-digit integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula