A064692 -id:A064692 - cp's OEIS frontend

A206439 Partial sums of A064692.

1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 7, 7, 7, 7, 8, 8, 9, 9, 11, 12, 13, 13, 13, 13, 14, 14, 15, 15, 17, 18, 19, 19, 19, 19, 20, 20, 21, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 35, 38, 40, 41, 41, 41, 41, 42, 42, 43, 43, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 60, 62, 63, 63, 63, 63, 64, 64, 65, 65, 67, 68, 71, 73, 75, 77, 80, 82, 85, 87, 91

Offset: 0

N. J. A. Sloane, Feb 27 2012

Cf. A064692.

Table[Sum[DigitCount[x].{0,0,0,1,0,1,0,2,1,1},{x,0,m}],{m,0,88}] (* Zak Seidov, Jul 25 2015 *)

A064532 Total number of holes in decimal expansion of the number n, assuming 4 has no hole.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 07 2001

Assumes that 4 is represented without a hole.

			8 has two holes so a(8) = 2.

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

Cf. A064529, A064530. Equals A064531 - 1.

Cf. A358439 (sum by number of digits).

a[n_ /; 0 <= n <= 9] := a[n] = {1, 0, 0, 0, 0, 0, 1, 0, 2, 1}[[n + 1]]; a[n_] := Total[a[#] + 1 &  /@ (id = IntegerDigits[n])] - Length[id];  Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 22 2013 *)
Table[DigitCount[x].{0, 0, 0, 0, 0, 1, 0, 2, 1, 1}, {x, 0, 104}] (* Michael De Vlieger, Feb 02 2017, after Zak Seidov at A064692 *)

h(n) = [1, 0, 0, 0, 0, 0, 1, 0, 2, 1][n];
a(n) = if (n, my(d=digits(n)); sum(i=1, #d, h(d[i]+1)), 1); \\ Michel Marcus, Nov 11 2022

def A064532(n):
    x=str(n)
    return x.count("0")+x.count("6")+x.count("8")*2+x.count("9") # Indranil Ghosh, Feb 02 2017

a(10i+j) = a(i) + a(j), etc.

More terms from Matthew Conroy, Oct 09 2001

A208568 Number of holes in n-th lower case letter of English alphabet (in hand-written form using a cursive script).

Original entry on oeis.org

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

Offset: 1

Alex Bellos and N. J. A. Sloane, Feb 28 2012, corrected Mar 01 2012

			a has one hole, b has one hole, c has no holes, ... (though the sequence refers to hand-written letters, not these printed letters).

Index entries for sequences related to holes in letters

Cf. A185374, A064530, A064692.

A327820 Smallest prime with n holes in its decimal digits.

Original entry on oeis.org

2, 19, 83, 89, 809, 1889, 4889, 46889, 48889, 468889, 688889, 3888889, 4888889, 28888889, 88884889, 288888889, 808888889, 4488888889, 8688888889, 48808888889, 48888888889, 288888888889, 888088888889, 1888888888889, 4888888888889, 48808888888889, 88848888888889

Offset: 0

Andrew Heathwaite, Sep 26 2019

Smallest prime p such that A064692(p) = n. Also record-holders in A327462. - Felix Fröhlich, Sep 27 2019

The sequence is not monotonically increasing: a(32) > a(33). - Giovanni Resta, Sep 27 2019

Giovanni Resta, Table of n, a(n) for n = 0..100

Cf. A064692, A249572, A250256, A327462.

s[0] = {1,2,3,5,7}; s[1] = {0,4,6,9}; 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, Sep 27 2019 *)

count_holes(n) = my(d=digits(n), i=0); for(k=1, #d, if(d[k]==0 || d[k]==4 || d[k]==6 || d[k]==9, i++, if(d[k]==8, i+=2))); i
a(n) = forprime(p=1, , if(count_holes(p)==n, return(p))) \\ Felix Fröhlich, Sep 27 2019

a(7) corrected and more terms added by Felix Fröhlich, Sep 27 2019

More terms from Giovanni Resta, Sep 27 2019

A185374 Number of holes in n-th lower case letter of English alphabet (in printed form using standard Times New Roman font).

Original entry on oeis.org

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

Offset: 1

Alex Bellos, Feb 29 2012

The letter a has one hole, b has one hole, c has no holes, ...

Index entries for sequences related to holes in letters

Cf. A208568, A064530, A064692.

A327462 Number of holes in decimal expansion of n-th prime.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 27 2019

This is A064692 restricted to the primes.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A035103, A064692, A249572, A250256, A327820.

forprime (p=2, 439, print1 (vecsum(apply(d -> [1, 0, 0, 0, 1, 0, 1, 0, 2, 1][1+d], digits(p))) ", ")) \\ Rémy Sigrist, Sep 27 2019

A064693 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, 2, 1, 2, 1, 3, 2, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 3, 3, 3, 4, 3, 4, 3, 5, 4, 3, 2, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 2, 2, 3

Offset: 0

Matthew Conroy, Oct 11 2001

			We assume 1,2,3,5 have no hole; 0,4,6,9 have 1 hole; 8 has two holes. So cutting 8 from a piece of paper creates three connected components: one for each hole and one for the remainder of the paper. Hence a(8)=3.

Cf. A064531. Equals A064692 + 1.

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

cp's OEIS Frontend

A206439 Partial sums of A064692.

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A064532 Total number of holes in decimal expansion of the number n, assuming 4 has no hole.

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A208568 Number of holes in n-th lower case letter of English alphabet (in hand-written form using a cursive script).

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A327820 Smallest prime with n holes in its decimal digits.

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A185374 Number of holes in n-th lower case letter of English alphabet (in printed form using standard Times New Roman font).

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A327462 Number of holes in decimal expansion of n-th prime.

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A064693 Number of connected components remaining when decimal expansion of the number n is cut from a piece of paper.

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

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Python