A093141 -id:A093141 - cp's OEIS frontend

A169967 Numbers whose decimal expansion contains only 0's and 4's.

0, 4, 40, 44, 400, 404, 440, 444, 4000, 4004, 4040, 4044, 4400, 4404, 4440, 4444, 40000, 40004, 40040, 40044, 40400, 40404, 40440, 40444, 44000, 44004, 44040, 44044, 44400, 44404, 44440, 44444, 400000, 400004, 400040, 400044, 400400, 400404, 400440, 400444

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A007088, A014263, A093141, A169964, A169965, A169966, A078243, A204093, A204094, A204095, A097256.

a169967 n = a169967_list !! (n-1)
a169967_list = map (* 4) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

FromDigits/@Tuples[{0,4},6] (* Harvey P. Dale, Dec 21 2018 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*4)))) \\ Charles R Greathouse IV, Nov 16 2011

a(n+1) = Sum_{k>=0} A030308(n,k)*A093141(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 4 * A007088(n-1).

A352156 Numbers m such that the smallest digit in the decimal expansion of 1/m is 2, ignoring leading and trailing 0's.

Original entry on oeis.org

4, 5, 16, 36, 40, 44, 45, 50, 108, 160, 216, 252, 288, 292, 308, 360, 364, 375, 396, 400, 404, 440, 444, 450, 500, 1024, 1080, 1375, 1600, 2072, 2160, 2368, 2520, 2880, 2920, 3080, 3125, 3375, 3600, 3640, 3750, 3848, 3960, 4000, 4040, 4125, 4224, 4368, 4400, 4440, 4500, 5000

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 19 2022

Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 (A003592) would be terms.
If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.

			m = 16 is a term since 1/16 = 0.0625 and the smallest term after the leading 0 is 2.
m = 216 is a term since 1/216 = 0.004629629629... and the smallest term after the leading 0's is 2.
m = 4444 is not a term since 1/4444 = 0.00022502250225... and the smallest term after the leading 0's is 0.

Cf. A341383.
Subsequences: A093141 \ {1}, A093143 \ {1}.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), this sequence (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 2 &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A352156_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,n), multiplicity(5,n)
        k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if s == '0' and min(str(c)) == '2':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '2':
                yield n
A352156_list = list(islice(A352156_gen(),20)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 2.

A337816 Numbers that can be written as (m * sum of digits of m) for some m.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 22, 25, 36, 40, 49, 52, 63, 64, 70, 81, 88, 90, 100, 112, 115, 124, 136, 144, 160, 162, 175, 190, 198, 202, 205, 208, 220, 238, 243, 250, 252, 280, 301, 306, 319, 324, 333, 352, 360, 364, 370, 400, 405, 412, 418, 424, 427, 448, 460, 468, 484, 486, 490

Offset: 1

Bernard Schott, Sep 23 2020

If 3 divides a(n), then 9 divides a(n).

			10 = 10 * (1+0);
22 = 11 * (1+1).

Range of A057147 and of A117570.
Similar sequences: A176995 (m + sum of digits of m), A336826 (m * product of digits of m), A337718 (m + product of digits of m).
Cf. A337817.
Some subsequences: A011557, A052268, A093141.

m = 500; Select[Union @ Table[k * Plus @@ IntegerDigits[k], {k, 0, m}], # <= m &] (* Amiram Eldar, Sep 23 2020 *)

is(k)={if(k==0, return(1)); fordiv(k, d, if(d*sumdigits(d)==k, return(1))); 0} \\ Andrew Howroyd, Sep 23 2020

A093140 Expansion of (1-6x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 5, 45, 445, 4445, 44445, 444445, 4444445, 44444445, 444444445, 4444444445, 44444444445, 444444444445, 4444444444445, 44444444444445, 444444444444445, 4444444444444445, 44444444444444445, 444444444444444445, 4444444444444444445, 44444444444444444445, 444444444444444444445

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 4*A001045(3n)/3+(-1)^n. Partial sums of A093141. A convex combination of 10^n and 1. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1, 1+k, 1+11k, 1+111k, ... This is the case for k=4.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001045, A002477, A093141, A102807.

CoefficientList[Series[(1-6x)/((1-x)(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{11,-10},{1,5},30] (* or *) Join[{1},Table[FromDigits[PadLeft[{5},n,4]],{n,30}]] (* Harvey P. Dale, Dec 17 2022 *)

G.f.: (1-6*x)/((1-x)*(1-10*x)).
a(n) = 4*10^n/9 + 5/9.
a(n+1) = (A102807(n+1)-A002477(n))/((Sum_{i=1..n} 2*10^i) + 3). [Roger L. Bagula, May 22 2010]
a(n) = 10*a(n-1)-5 with a(0)=1. - Vincenzo Librandi, Aug 02 2010
a(n) = 11*a(n-1)-10*a(n-2). - Wesley Ivan Hurt, May 20 2021
E.g.f.: exp(x)*(4*exp(9*x) + 5)/9. - Elmo R. Oliveira, Aug 17 2024

a(19)-a(22) from Elmo R. Oliveira, Aug 17 2024

A346178 Expansion of (1-2x)/(1-10x).

Original entry on oeis.org

1, 8, 80, 800, 8000, 80000, 800000, 8000000, 80000000, 800000000, 8000000000, 80000000000, 800000000000, 8000000000000, 80000000000000, 800000000000000, 8000000000000000, 80000000000000000, 800000000000000000, 8000000000000000000, 80000000000000000000

Offset: 0

Felix Fröhlich, Jul 09 2021

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

Cf. expansion of (1-k*x)/(1-10*x) A011557 (k=0), A196662 (k=3), A090019 (k=4), A093143 (k=5), A093141 (k=6), A093138 (k=7), A093136 (k=8).

PARI
```
Vec((1-2*x)/(1-10*x) + O(x^20))
```

a(n) = 8*10^(n-1), n>0.

E.g.f.: (1 + 4*exp(10*x))/5. - Stefano Spezia, Jul 09 2021

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A169967 Numbers whose decimal expansion contains only 0's and 4's.

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A352156 Numbers m such that the smallest digit in the decimal expansion of 1/m is 2, ignoring leading and trailing 0's.

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A337816 Numbers that can be written as (m * sum of digits of m) for some m.

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A093140 Expansion of (1-6*x)/((1-x)*(1-10*x)).

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A346178 Expansion of (1-2*x)/(1-10*x).

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A093140 Expansion of (1-6x)/((1-x)(1-10*x)).

A346178 Expansion of (1-2x)/(1-10x).