A367728 -id:A367728 - cp's OEIS frontend

A367593 Nonnegative integers k such that (R(k) - 1)/(k + 1) is an integer, where R(k) is the digit reversal of k.

0, 1, 10, 100, 147, 1000, 1099, 1407, 10000, 14007, 100000, 140007, 1000000, 1400007, 2124736, 10000000, 14000007, 100000000, 123456789, 140000007, 1000000000, 1234506789, 1400000007, 10000000000, 12345006789, 14000000007, 21247524736, 100000000000, 123450006789

Offset: 1

Stefano Spezia, Nov 24 2023

Among the terms of this sequence, there are:
  the powers of 10 (cf. A011557);
  the numbers of the form 14*10^h + 7 = 7*A199682(h) for h > 0;
  the numbers of the form 12345*10^m + 6789 = A367650(m) for m > 3.
From Chai Wah Wu, Dec 01 2023: (Start)
First digit of positive terms must be either 1 or 2. If first digit is 1, then last digit must be 0,1,4,5,7 or 9. If first digit is 2, then last digit is 6. In particular, if a, b, c are the first digit, last digit and (R(k)-1)/(k+1) of a term k>0, then (a, b, c) must take on values from one of the following triples:
  (a, b, c): (1, 0, 0), (1, 1, 0), (1, 4, 2), (1, 4, 4), (1, 5, 5), (1, 7, 5),
  (1, 9, 4), (1, 9, 5), (1, 9, 6), (1, 9, 7), (1, 9, 8), (1, 9, 9), (2, 6, 3).
Numbers of the form 21 [2475]* 24736 are terms of this sequence, where [2475]* denote a (possibly zero) repetition of the digits 2475. The first few terms of this form are: 2124736, 21247524736, 212475247524736, ...
Similarly numbers of the form 21261 [2475]* 24738736 are terms of this sequence:
  2126124738736, 21261247524738736, 212612475247524738736, ... (End)
More generally, numbers of the form 21 [261]^k [2475]* 2473 [873]^k 6 are terms of this sequence, where [261]^k denote the digits '261' repeated k times with k>=0: e.g. 21261261247524752475247524738738736, ... It appears that all terms with first digit 2 and last digit 6 are of this form. - Chai Wah Wu, Dec 02 2023

			123456789 is a term since (987654321 - 1)/(123456789 + 1) = 8, which is an integer.

Chai Wah Wu, Table of n, a(n) for n = 1..44

Cf. A004086, A011557, A199682, A367650.

Cf. A367727, A367728, A367740.

a={}; For[k=0, k<=10^10, k++,If[IntegerQ[(FromDigits[Reverse[IntegerDigits[k]]]-1)/(k+1)],AppendTo[a,k]]]; a
Select[Range[0,10^6],IntegerQ[(IntegerReverse[#]-1)/(#+1)]&] (* The program generates the first 13 terms of the sequence. *) (* Harvey P. Dale, Aug 11 2024 *)

isok(k) = denominator((fromdigits(Vecrev(digits(k))) - 1)/(k + 1)) == 1; \\ Michel Marcus, Nov 30 2023

def digit_reversal(n):
    return int(str(n)[::-1])
def find_integers():
    result = []
    for k in range(0, 10**10):
        reversed_k = digit_reversal(k)
        if (reversed_k - 1) % (k + 1) == 0:
            result.append(k)
    return result
integers_list = find_integers()
print(integers_list)

from itertools import product, count, islice
def A367593_gen(): # generator of terms
    yield from (0,1,10)
    for l in count(1):
        m = 10**(l+1)
        for d in product('0123456789',repeat=l):
            for a, b, c in ((1, 0, 0), (1, 1, 0), (1, 4, 2), (1, 5, 5), (1, 7, 5)):
                k = a*m+int(s:=''.join(d))*10+b
                r = b*m+int(s[::-1])*10+a
                if c*(k+1)==r-1:
                    yield k
            a,b = 1,9
            k = a*m+int(s:=''.join(d))*10+b
            r = b*m+int(s[::-1])*10+a
            if not (r-1)%(k+1):
                yield k
        a,b,c=2,6,3
        for d in product('0123456789',repeat=l):
            k = a*m+int(s:=''.join(d))*10+b
            r = b*m+int(s[::-1])*10+a
            if c*(k+1)==r-1:
                yield k
A367593_list = list(islice(A367593_gen(),20)) # Chai Wah Wu, Dec 01 2023

A367728(a(n)) = 1.

A367727 a(n) is the numerator of (R(n) - 1)/(n + 1), where R(n) is the digit reversal of n.

Original entry on oeis.org

-1, 0, 1, 1, 3, 2, 5, 3, 7, 4, 0, 5, 20, 15, 8, 25, 60, 35, 80, 9, 1, 1, 21, 31, 41, 51, 61, 71, 81, 91, 2, 3, 2, 16, 6, 13, 62, 36, 82, 23, 3, 13, 23, 3, 43, 53, 63, 73, 83, 93, 4, 7, 24, 17, 4, 27, 64, 37, 84, 47, 5, 15, 25, 35, 9, 5, 65, 75, 85, 19, 6, 2, 26

Offset: 0

Stefano Spezia, Nov 28 2023

Stefano Spezia, Table of n, a(n) for n = 0..10000

Cf. A004086, A367593, A367728 (denominator).

a[n_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-1)/(n+1)]; Array[a,73,0]

a(n) = numerator((fromdigits(Vecrev(digits(n)))-1)/(n+1)); \\ Michel Marcus, Nov 28 2023

A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 3, 1, 3, 3, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 4, 2, 4, 1, 4, 2, 4, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 4, 6, 12, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367728.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the denominators begins:
  1, 1, 1, 1,  1,  1,  1,  1, ...
  1, 1, 3, 2,  5,  3,  7,  4, ...
  1, 3, 1, 5,  3,  7,  2,  9, ...
  1, 2, 5, 1,  7,  4,  3,  5, ...
  1, 5, 3, 7,  1,  9,  5, 11, ...
  1, 3, 7, 4,  9,  1, 11,  6, ...
  1, 7, 2, 3,  5, 11,  1, 13, ...
  1, 4, 9, 5, 11,  6, 13,  1, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367824 (numerator), A367827 (antidiagonal sums).

Cf. A000012 (n=0), A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106615, A106617, A231190, A367728 (k=1).

A[0,0]=1; A[n_,k_]:=Denominator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(1, n) = A026741(n+1).
A(2, n) = A060819(n+2).
A(3, n) = A060789(n+3).
A(4, n) = A106609(n+4).
A(5, n) = A106611(n+5).
A(6, n) = A051724(n+6).
A(7, n) = A106615(n+7).
A(8, n) = A106617(n+8) = A231190(n+16).
A(9, n) = A106619(n+9).
A(10, n) = A106612(n+10).

A367740 a(n) = A367727(A367593(n)).

Original entry on oeis.org

-1, 0, 0, 0, 5, 0, 9, 5, 0, 5, 0, 5, 0, 5, 3, 0, 5, 0, 8, 5, 0, 8, 5, 0, 8, 5, 3, 0, 8, 5, 0, 8, 5, 3, 0, 8, 5, 0, 8, 5, 3, 0, 8, 5

Offset: 1

Stefano Spezia, Nov 29 2023

Restricted to integers k where k + 1 divides R(k) - 1 the terms are (R(k) - 1) / (k + 1), where R(k) denotes the digit reversal of k. - Peter Luschny, Dec 01 2023

The 0's correspond to the powers of 10 (cf. A011557), the 5's correspond to the numbers of the form 14*10^h + 7 = 7*A199682(h) for h > 0, and the 8's correspond to the numbers of the form 12345*10^m + 6789 = A367650(m) for m > 3. The only negative term -1 corresponds to A367593(1) = 0. - Stefano Spezia, Dec 01 2023

Cf. A367593, A367727, A367728.

Cf. A004086, A011557, A199682, A367650.

def A367740List(upto):
    L = []
    for n in range(upto):
        rev = int(str(n)[::-1])
        if (rev - 1) % (n + 1) == 0:
            L.append((rev - 1) // (n + 1))
    return L
print(A367740List(10**6))  # Peter Luschny, Dec 01 2023

from itertools import product, count, islice
def A367740_gen(): # generator of terms
    yield from (-1,0,0)
    for l in count(1):
        m = 10**(l+1)
        for d in product('0123456789',repeat=l):
            for a, b, c in ((1, 0, 0), (1, 1, 0), (1, 4, 2), (1, 5, 5), (1, 7, 5)):
                k = a*m+int(s:=''.join(d))*10+b
                r = b*m+int(s[::-1])*10+a
                if c*(k+1)==r-1:
                    yield c
            a,b = 1,9
            k = a*m+int(s:=''.join(d))*10+b
            r = b*m+int(s[::-1])*10+a
            p,q = divmod(r-1,k+1)
            if not q:
                yield p
        a,b,c=2,6,3
        for d in product('0123456789',repeat=l):
            k = a*m+int(s:=''.join(d))*10+b
            r = b*m+int(s[::-1])*10+a
            if c*(k+1)==r-1:
                yield c
A367740_list = list(islice(A367740_gen(),20)) # Chai Wah Wu, Dec 01 2023

-1 <= a(n) <= 9.

a(30)-a(44) from Chai Wah Wu, Dec 02 2023

cp's OEIS Frontend

A367593 Nonnegative integers k such that (R(k) - 1)/(k + 1) is an integer, where R(k) is the digit reversal of k.

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A367727 a(n) is the numerator of (R(n) - 1)/(n + 1), where R(n) is the digit reversal of n.

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A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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A367740 a(n) = A367727(A367593(n)).

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