A021499 -id:A021499 - cp's OEIS frontend

A341383 Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

5, 45, 50, 450, 495, 500, 819, 825, 4500, 4545, 4950, 4995, 5000, 8190, 8250, 8325, 45000, 45045, 45450, 47619, 49500, 49950, 49995, 50000, 81819, 81900, 82500, 83250, 83325, 89109, 450000, 450045, 450450, 454500, 454545, 476190, 495000, 499500, 499950, 499995, 500000

Offset: 1

Bernard Schott, Feb 10 2021

If m is a term, 10*m is also a term.
5 is the only prime up to 2.6*10^8 (comments in A333237).
Some subsequences: {45, 4545, 454545, ...}, {45045, 45045045, 45045045045, ...}, {45, 495, 4995, 49995, ...}, {819, 81819, 8181819, ...}, {825, 8325, 83325, 833325...}, ...
The subsequence of terms where 1/m has only digits {0,2} is m = 5*A333402 = 5, 45, 50, etc.  A333402 is those t where 1/t has only digits {0,1}, so that 1/(5*t) = 2*(1/t)*(1/10) has digits {0,2}, starting from 1/5 = 0.2.  These m are also A333402/2 of the even terms from A333402, since A333402 (like here) is self-similar in that the multiples of 10, divided by 10, are the sequence itself. - Kevin Ryde, Feb 13 2021

			As 1/45 = 0.0202020202..., 45 is a term.
As 1/825 = 0.0012121212121212...., 825 is a term.
As 1/47619 = 0.000021000021000021..., 47619 is a term.
As 1/4545045 = 0.000000220019824..., 4545045 is not a term.

Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Similar with largest digit k: A333402 (k=1), A333237 (k=9).
Subsequence: A093143 \ {1}.
Decimal expansion: A021499 (1/495), A021823 (1/819).

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 2 &] (* Amiram Eldar, Feb 10 2021 *)

from itertools import count, islice
from sympy import n_order, multiplicity
def A341383_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,m), multiplicity(5,m)
        if max(str(10**(max(m2,m5)+n_order(10,m//2**m2//5**m5))//m)) == '2':
            yield m
A341383_list = list(islice(A341383_gen(),10)) # Chai Wah Wu, Feb 07 2022

Missing terms added by Amiram Eldar, Feb 10 2021

A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, 0, -2, 0, 3, 2, -2, -3, 0, 4, 5, 0, -5, -4, 0, 5, 9, 5, -5, -9, -5, 0, 6, 14, 14, 0, -14, -14, -6, 0, 7, 20, 28, 14, -14, -28, -20, -7, 0, 8, 27, 48, 42, 0, -42, -48, -27, -8, 0, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 0, 10, 44, 110

Offset: 0

Reinhard Zumkeller, Jul 18 2015

sign

A214292 gives first differences per row in Pascal's triangle.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A014473, A163866, A214292.

Cf. A000108, A021499, A047171, A074331.

a259525 n = a259525_list !! n
a259525_list = zipWith (-) (tail pascal) pascal
                           where pascal = concat a007318_tabl

[k eq n select 0 else (n-2*k-1)*Binomial(n,k+1)/(n-k): k in [0..n], n in [0..14]]; // G. C. Greubel, Apr 25 2024

Table[If[k==n, 0, ((n-2*k-1)/(n-k))*Binomial[n,k+1]], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Apr 25 2024 *)

flatten([[binomial(n,k+1) -binomial(n,k) +int(k==n) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Apr 25 2024

From G. C. Greubel, Apr 25 2024: (Start)
If viewed as a triangle then:
T(n, k) = binomial(n, k+1) - binomial(n, k), with T(n, n) = 0.
T(n, n-k) = - T(n, k), for 0 <= k < n.
T(2*n, n) = [n=0] - A000108(n).
Sum_{k=0..n} T(n, k) = 0 (row sums).
Sum_{k=0..floor(n/2)} T(n, k) = A047171(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A021499(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A074331(n-1). (End)

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A341383 Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

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A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.

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