A367593 -id:A367593 - cp's OEIS frontend

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

-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

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

A367728 a(n) is the denominator of (R(n) - 1)/(n + 1), where R(n) is the digit reversal of n.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 1, 6, 13, 7, 3, 8, 17, 9, 19, 2, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 8, 3, 17, 5, 9, 37, 19, 39, 10, 41, 42, 43, 4, 45, 46, 47, 48, 49, 50, 51, 26, 53, 27, 5, 28, 57, 29, 59, 30, 61, 62, 63, 64, 13, 6, 67, 68, 69, 14, 71, 9

Offset: 0

Stefano Spezia, Nov 28 2023

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

Cf. A004086, A367593, A367727 (numerator).

a[n_]:=Denominator[(FromDigits[Reverse[IntegerDigits[n]]]-1)/(n+1)]; Array[a,72,0]

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

A367650 a(n) = 12345*10^n + 6789.

Original entry on oeis.org

19134, 130239, 1241289, 12351789, 123456789, 1234506789, 12345006789, 123450006789, 1234500006789, 12345000006789, 123450000006789, 1234500000006789, 12345000000006789, 123450000000006789, 1234500000000006789, 12345000000000006789, 123450000000000006789, 1234500000000000006789

Offset: 0

Stefano Spezia, Nov 25 2023

For n > 3, a(n) satisfies (R(a(n)) - 1)/(a(n) + 1) = 8, where R(x) = A004086(x) is decimal digit reversal.

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

Cf. A004086, A011557, A367593.

LinearRecurrence[{11,-10},{19134,130239},20]

a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
O.g.f.: 9*(2126 - 8915*x)/((1 - x)*(1 - 10*x)).
E.g.f.: 3*exp(x)*(2263 + 4115*exp(9*x)).

cp's OEIS Frontend

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

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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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A367728 a(n) is the denominator of (R(n) - 1)/(n + 1), where R(n) is the digit reversal of n.

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A367650 a(n) = 12345*10^n + 6789.

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