A001745 -id:A001745 - cp's OEIS frontend

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.

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

Offset: 0

Rick L. Shepherd, Nov 01 2014

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.

			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      .
.         XX  XX      .       .    XX       XX      .
.        XX   XX      .       .    XX       XX      .
.       XX    XX      .       .    XX       XX      .
.      XX     XX      .       .    XX       XX      .
.     XX      XX      .       .    XX       XX      .
.    XX       XX      .       .    XX       XX      .
.    XXXXXXXXXXXXX    .       .    XXXXXXXXXXXXX    .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.             XX      .       .             XX      .
.                     .       .                     .
.      "Holey" 4      .       .    "Non-holey" 4    .
. . . . . . . . . . . .       . . . . . . . . . . . . (End)

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).

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

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

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

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

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

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)
a(n) = (2/9)*(10^(n/2)*(4*((n+1) mod 2) + 11*sqrt(2/5)*(n mod 2)) - 4) for n >= 1. - Alan Michael Gómez Calderón, May 04 2025

Offset corrected by Brady Haran, Nov 27 2018

A250256 Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Nov 15 2014

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

			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.

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).

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

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

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 *)

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)
E.g.f.: (9 + 45*x - 40*cosh(x) + 31*cosh(sqrt(10)*x) - 40*sinh(x) + 4*sqrt(10)*sinh(sqrt(10)*x))/45. - Stefano Spezia, Aug 11 2025

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

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Nov 15 2014

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.

			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.

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

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

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

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 *)

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)

A250258 Least nonnegative integer whose decimal digits divide the plane into n regions (A250257 variant).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Nov 15 2014

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.

			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.

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

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

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

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 *)

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)

A261449 Prime numbers whose decimal digits contain a total of two loops.

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Aug 19 2015

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

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

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

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

More terms from Giovanni Resta, Aug 19 2015

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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.

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A250256 Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

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

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A250258 Least nonnegative integer whose decimal digits divide the plane into n regions (A250257 variant).

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Magma

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A261449 Prime numbers whose decimal digits contain a total of two loops.

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