A249572 -id:A249572 - cp's OEIS frontend

A281191 Number of holes in the (American) English name of n (as printed in lower case).

2, 2, 1, 2, 1, 1, 0, 2, 3, 1, 1, 3, 2, 2, 3, 2, 2, 4, 5, 3, 1, 3, 2, 3, 2, 2, 1, 3, 4, 2, 0, 2, 1, 2, 1, 1, 0, 2, 3, 1, 1, 3, 2, 3, 2, 2, 1, 3, 4, 2, 0, 2, 1, 2, 1, 1, 0, 2, 3, 1, 0, 2, 1, 2, 1, 1, 0, 2, 3, 1, 2, 4, 3, 4, 3, 3, 2, 4, 5, 3, 3, 5, 4, 5, 4, 4, 3, 5, 6, 4, 1, 3, 2, 3, 2, 2, 1, 3, 4, 2, 5, 7, 6, 7, 6

Offset: 0

Rick L. Shepherd, Jan 16 2017

For this sequence a font is used where a, b, d, e, o, p, and q each have one hole, g has two, and all other letters have no holes.

			The term a(101) = 7 because the name "one hundred one" contains seven total holes in these letters: o, e, d, e, d, o, and e.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005589, A249572.

a:= n-> (s-> add((t-> `if`(t in {"a", "b", "d", "e", "o", "p", "q"}, 1,
        `if`(t="g", 2, 0)))(s[i]), i=1..length(s)))(convert(n, english)):
seq(a(n), n=0..104);  # Alois P. Heinz, Jul 30 2023

A337099 Largest positive number using exactly n segments on a calculator display (when '6' and '7' are represented using 6 resp. 3 segments).

Original entry on oeis.org

1, 7, 11, 71, 111, 711, 1111, 7111, 11111, 71111, 111111, 711111, 1111111, 7111111, 11111111, 71111111, 111111111, 711111111, 1111111111, 7111111111, 11111111111, 71111111111, 111111111111, 711111111111, 1111111111111, 7111111111111, 11111111111111, 71111111111111

Offset: 2

Suren Suren, Sep 29 2020

The sequence begins with a(2) = 1 since at least two segments are needed to form any digit. It requires two segments to form the digit 1 and three segments to form the digit 7.

All other digits use more than 3 segments.

Index entries for sequences related to calculator display
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A063720 (number of segments), A216261 (smallest number), A249572.

CoefficientList[Series[(1 + 6*x - 6*x^2)/(1 - x - 10*x^2 + 10*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Nov 07 2020 *)

a(n+2) = 10*a(n) + 1 for n >= 2.
a(2*n) = (10^n - 1)/9 ; a(2*n + 1) = ((10^n - 1)/9) + 6*10^(n - 1).
From Stefano Spezia, Sep 29 2020: (Start)
G.f.: x^2*(1 + 6*x - 6*x^2)/(1 - x - 10*x^2 + 10*x^3).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 4. (End)

More terms from Stefano Spezia, Sep 29 2020

A363377 Largest positive integer having n holes that can be made using the fewest possible digits.

Original entry on oeis.org

7, 9, 8, 98, 88, 988, 888, 9888, 8888, 98888, 88888, 988888, 888888, 9888888, 8888888, 98888888, 88888888, 988888888, 888888888, 9888888888, 8888888888, 98888888888, 88888888888, 988888888888, 888888888888, 9888888888888, 8888888888888, 98888888888888, 88888888888888, 988888888888888

Offset: 0

Julia Zimmerman, May 29 2023

Each decimal digit has 0, 1 or 2 holes so that n holes requires A065033(n) digits.

			For n=0, the largest integer with no holes in it that is as short as possible is 7 (9 is larger, but has 1 hole; 11 is larger and has no holes, but is longer at length 2 > length 1).
For n=1, the largest integer with 1 hole that is as short as possible is 9 (following the same kind of reasoning as with n=0).

Index entries for sequences related to holes in digits.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A002281 and A002282 (number of holes), A065033 (digits required).
Cf. A249572 and A250256 (smallest number).
Cf. A337099 (largest 7-segment).

CoefficientList[Series[(7 + 2 x - 71 x^2 + 70 x^3)/((1 - x) (1 - 10 x^2)), {x, 0, 30}], x] (* Michael De Vlieger, Jul 05 2023 *)

A363377=lambda n: (8+n%2*81)*10**(n>>1)//9 if n else 7
print([A363377(n) for n in range(30)]) # Natalia L. Skirrow, Jun 26 2023

From Natalia L. Skirrow, Jun 26 2023: (Start)
a(n) = (89*(10^((n-1)/2))-8)/9 for odd n; a(n) = 8*(10^(n/2)-1)/9 for even n >= 2.
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3), for n >= 4.
G.f.: (7+2*x-71*x^2+70*x^3)/((1-x)*(1-10*x^2)).
E.g.f.: (80*cosh(sqrt(10)*x) + 89*sqrt(10)*sinh(sqrt(10)*x) - 80*e^x)/90 + 7. (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

A374349 Integers >=0 whose decimal digits are topologically distinct from those of any smaller number.

Original entry on oeis.org

0, 1, 8, 10, 11, 18, 40, 48, 88, 100, 101, 108, 111, 118, 188, 400, 408, 488, 888, 1000, 1001, 1008, 1011, 1018, 1088, 1111, 1118, 1188, 1888, 4000, 4008, 4088, 4888, 8888, 10000, 10001, 10008, 10011, 10018, 10088, 10111, 10118, 10188, 10888, 11111, 11118

Offset: 1

Charles L. Hohn, Jul 05 2024

Assumes 0 without a slash or a center dot, closed 4, 6, and 9, and no overlapping of multiple digits. Digits homologous to a (flattened) sphere: 1, 2, 3, 5, 7; to a torus: 0, 4, 6, 9; to a double torus: 8. Sequence is a run of the terms in ascending numeric order.

All topologically distinct terms can be represented by nondecreasing sequences of strings of 0s, 1s, and 8s. However, terms cannot begin with 0. Therefore, if a string has 0s, then (i) if there are any 1s, one of them moves to the front, (ii) else, the first 0 is replaced with 4. Sequence is the resulting strings sorted as base-10 numbers. - Michael S. Branicky, Jul 11 2024

			0 is homologous to 1 torus, so a(1)=0.
1 is homologous to 1 sphere, so a(2)=1.
2 is homologous to 1 sphere, same as 1, so it is not in the sequence.
4 is homologous to 1 torus, same as 0, so it is not in the sequence.
8 is homologous to 1 double torus, so a(3)=8.
10 is homologous to 1 sphere and 1 torus, so a(4)=10.
11 is homologous to 2 spheres, so a(5)=11.
14 is homologous to 1 sphere and 1 torus, same as 10, so it is not in the sequence.
41 is homologous to 1 sphere and 1 torus, same as 10, so it is not in the sequence.

Cf. A179239, A064692, A249572.

df(d, c)=(10^c-1)/9*d
n=0; a=0; at=1; while(true, a++; at+=a+1; ac=0; for(b=0, a, for(c=0, b, n++; print(n, " ", if(n<=2, n-1, ac+b-c+1

from itertools import count, islice, combinations_with_replacement as cwr
def agen(): # generator of terms
    after = {"1":"018", "4":"08", "8":"8"}
    yield from (0, 1, 8)
    for digits in count(2):
        for first in "148":
            for rest in cwr(after[first], digits-1):
                yield int(first + "".join(rest))
print(list(islice(agen(), 50))) # Michael S. Branicky, Jul 07 2024

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A281191 Number of holes in the (American) English name of n (as printed in lower case).

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A337099 Largest positive number using exactly n segments on a calculator display (when '6' and '7' are represented using 6 resp. 3 segments).

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A363377 Largest positive integer having n holes that can be made using the fewest possible digits.

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

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A374349 Integers >=0 whose decimal digits are topologically distinct from those of any smaller number.

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