A001745 -id:A001745

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A249572

Least positive integer whose decimal digits divide the plane into n+1 regions. Equivalently, least positive integer with n holes in its decimal digits.

+10
16

1, 4, 8, 48, 88, 488, 888, 4888, 8888, 48888, 88888, 488888, 888888, 4888888, 8888888, 48888888, 88888888, 488888888, 888888888, 4888888888, 8888888888, 48888888888, 88888888888, 488888888888, 888888888888, 4888888888888, 8888888888888, 48888888888888

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Leading zeros are not permitted. Variations are possible depending upon whether 4 is considered "holey" (if not, replace each "4" with a "6") and whether nonnegative integers are permitted (a(2) becomes 0). In each case, all terms after the first could be considered "wholly holey," as could all terms of A001743 and A001744, as each digit contains a hole (loop). The analogous sequence of bits for base 2 is simply A011557, the powers of 10, read instead as binary numbers, i.e., as powers of two.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video.

Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, A blind spot for large language models: Supradiegetic linguistic information, arXiv:2306.06794 [cs.CL], 2023.

Index entries for sequences related to holes in digits

Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

FORMULA

a(n) = 10*a(n-2) + 8 for n >= 3.

From Chai Wah Wu, Dec 14 2016: (Start)

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 4.

G.f.: (10*x^3 - 6*x^2 + 3*x + 1)/((x - 1)*(10*x^2 - 1)). (End)

EXAMPLE

From Jon E. Schoenfield, Nov 15 2014: (Start)

This sequence uses "holey" fours. So a(1)=4, because

. . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. XXXX . . XX XX .

. XX XX . . XX XX .

. XXXXXXXXXXXXX . . XXXXXXXXXXXXX .

. XX . . XX .

. . . .

. "Holey" 4 . . "Non-holey" 4 .

. . . . . . . . . . . . . . . . . . . . . . . . (End)

MAPLE

a:= n-> `if`(n=0, 1, parse(cat(4*(irem(n, 2, 'q')), 8$q))):

seq(a(n), n=0..30); # Alois P. Heinz, Nov 01 2014

MATHEMATICA

LinearRecurrence[{1, 10, -10}, {1, 4, 8, 48}, 50]] (* Paolo Xausa, May 31 2023 *)

PROG

(Magma) I:=[1, 4, 8, 48]; [n le 4 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 17 2014

(PARI) A249572(n)=10^(n\2)*if(n%2, 45-(n>1)*5, 22)\45 \\ "(..., 9-(n>1), 4.4)\9" would be shorter but cause problems beyond realprecision. - M. F. Hasler, Jul 25 2015

CROSSREFS

Cf. A001743, A001744, A001745, A001746, A002282, A011557, A000079.

The analogous sequence using 6 instead of 4 is A250256. - N. J. A. Sloane, Sep 27 2019

KEYWORD

nonn,base,easy

AUTHOR

Rick L. Shepherd, Nov 01 2014

EXTENSIONS

Offset corrected by Brady Haran, Nov 27 2018

STATUS

approved

A250256

Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

+10
7

1, 6, 8, 68, 88, 688, 888, 6888, 8888, 68888, 88888, 688888, 888888, 6888888, 8888888, 68888888, 88888888, 688888888, 888888888, 6888888888, 8888888888, 68888888888, 88888888888, 688888888888, 888888888888, 6888888888888, 8888888888888, 68888888888888

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Equivalently, with offset 0, least positive integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A249572 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.

For n > 2, a(n) + a(n+1) divides the plane into 2 regions. For n > 1, a(2n) - a(2n-1) divides the plane into n+1 regions. For n >= 1, a(2n+1) - a(2n) divides the plane into n regions. - Ivan N. Ianakiev, Feb 23 2015

LINKS

Table of n, a(n) for n=1..28.

Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

FORMULA

a(n) = 10*a(n-2) + 8 for n >= 4.

From Chai Wah Wu, Jul 12 2016: (Start)

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 4.

G.f.: x*(10*x^3 - 8*x^2 + 5*x + 1)/((x - 1)*(10*x^2 - 1)). (End)

EXAMPLE

The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller positive integer does this, so a(4) = 68.

MATHEMATICA

Join[{1, 6, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

PROG

(Magma) I:=[1, 6, 8, 68]; [n le 4 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 15 2014

CROSSREFS

Cf. A249572, A250257, A250258, A001743, A001744, A001745, A001746, A002282.

KEYWORD

nonn,base,easy

AUTHOR

Rick L. Shepherd, Nov 15 2014

STATUS

approved

A250257

Least nonnegative integer whose decimal digits divide the plane into n regions.

+10
2

1, 0, 8, 48, 88, 488, 888, 4888, 8888, 48888, 88888, 488888, 888888, 4888888, 8888888, 48888888, 88888888, 488888888, 888888888, 4888888888, 8888888888, 48888888888, 88888888888, 488888888888, 888888888888, 4888888888888, 8888888888888, 48888888888888

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Equivalently, with offset 0, least nonnegative integer with n holes in its decimal digits. Leading zeros are not permitted. Identical to A249572 except that a(2) = 0, not 4. See also the comments in A249572.

LINKS

Table of n, a(n) for n=1..28.

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

FORMULA

a(n) = 10*a(n-2) + 8 for n >= 5.

From Chai Wah Wu, Jul 12 2016: (Start)

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 5.

G.f.: x*(-40*x^4 + 50*x^3 - 2*x^2 - x + 1)/((x - 1)*(10*x^2 - 1)). (End)

EXAMPLE

The integer 48, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller nonnegative integer does this, so a(4) = 48.

MATHEMATICA

Join[{1, 0, 8}, RecurrenceTable[{a[1]==48, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

PROG

(Magma) I:=[1, 0, 8, 48, 88]; [n le 5 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 15 2014

CROSSREFS

Cf. A249572, A250256, A250258, A001743, A001744, A001745, A001746, A002282.

KEYWORD

nonn,base,easy

AUTHOR

Rick L. Shepherd, Nov 15 2014

STATUS

approved

A250258

Least nonnegative integer whose decimal digits divide the plane into n regions (A250257 variant).

+10
2

1, 0, 8, 68, 88, 688, 888, 6888, 8888, 68888, 88888, 688888, 888888, 6888888, 8888888, 68888888, 88888888, 688888888, 888888888, 6888888888, 8888888888, 68888888888, 88888888888, 688888888888, 888888888888, 6888888888888, 8888888888888, 68888888888888

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Equivalently, with offset 0, least nonnegative integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A250257 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.

LINKS

Table of n, a(n) for n=1..28.

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

FORMULA

a(n) = 10*a(n-2) + 8 for n >= 5.

a(n) = A250256(n), n<>2.

From Chai Wah Wu, Jul 12 2016: (Start)

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 5.

G.f.: x*(-60*x^4 + 70*x^3 - 2*x^2 - x + 1)/((x - 1)*(10*x^2 - 1)). (End)

EXAMPLE

The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller nonnegative integer does this, so a(4) = 68.

MATHEMATICA

Join[{1, 0, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

PROG

(Magma) I:=[1, 0, 8, 68, 88]; [n le 5 select I[n] else 10*Self(n-2)+8: n in [1..40]]; // Vincenzo Librandi, Nov 16 2014

CROSSREFS

Cf. A249572, A250256, A250257, A001743, A001744, A001745, A001746, A002282.

KEYWORD

nonn,base,easy

AUTHOR

Rick L. Shepherd, Nov 15 2014

STATUS

approved

A261449

Prime numbers whose decimal digits contain a total of two loops.

+10
0

83, 109, 149, 181, 199, 269, 281, 283, 349, 383, 401, 419, 439, 443, 461, 463, 467, 479, 491, 509, 569, 587, 599, 601, 607, 619, 641, 643, 647, 659, 661, 691, 709, 769, 787, 811, 821, 823, 827, 853, 857, 877, 907, 919, 929, 941, 947, 967, 991, 997, 1019, 1039

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Of the digits, 0 through 9, {0, 4, 6, 9} have one loop, 8 has two loops, and all the rest have none. - Robert G. Wilson v, Aug 20 2015

LINKS

Table of n, a(n) for n=1..52.

EXAMPLE

83 is the first term of the sequence. The digit 8 contains two closed curves.

MATHEMATICA

Select[Prime@ Range@ 200, 2 == Total[{ 1, 0, 0, 0, 1, 0, 1, 0, 2, 1}[[1 + IntegerDigits@ #]]]&] (* Giovanni Resta, Aug 19 2015 *)

CROSSREFS

Cf. A001729, A190222, A001743, A001744, A001745, A249572.

KEYWORD

nonn,less,easy,base

AUTHOR

Altug Alkan, Aug 19 2015

EXTENSIONS

More terms from Giovanni Resta, Aug 19 2015

STATUS

approved

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