cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A078257 a(n) = denominator(N) where N = 0.123...n (concatenation of 1 to n after decimal point).

Original entry on oeis.org

10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, 1000000000, 10000000000, 10000000000000, 125000000000000, 100000000000000000, 5000000000000000000, 200000000000000000000, 25000000000000000000000, 10000000000000000000000000, 500000000000000000000000000, 100000000000000000000000000000
Offset: 1

Views

Author

Amarnath Murthy, Nov 24 2002

Keywords

Comments

Conjecture: sequence is not equal to the sequence of denominators presented in A172495 and A172506. - Jaroslav Krizek, Feb 05 2010
The conjecture is false for both other sequences; see A172495 and A172506 for proofs. - Michael S. Branicky, Nov 30 2022

Examples

			a(1) = 10 as 10*0.1 = 1, a(2) = 25 as 25*0.12 = 3.
		

Crossrefs

Cf. A058183, A078258 (numerators), A172495, A172506.

Programs

  • PARI
    a(n) = {my(s = ""); for (k=1, n, s = concat(s, Str(k))); denominator(eval(s)/10^(#s));} \\ Michel Marcus, Jan 15 2019
    
  • Python
    from itertools import count, islice
    def agen(): # generator of terms
        num, den, pow = 0, 1, 0
        for n in count(1):
            sn = str(n)
            num = num*10**len(sn) + n
            den *= 10**len(sn)
            pow += len(sn)
            nr, dr, c2, c5 = num, den, pow, pow
            while nr%2 == 0 and c2 > 0: nr //= 2; dr //= 2; c2 -= 1
            while nr%5 == 0 and c5 > 0: nr //= 5; dr //= 5; c5 -= 1
            yield dr
    print(list(islice(agen(), 19))) # Michael S. Branicky, Nov 30 2022

Formula

a(n) = denominator(Sum_{k=1..n} k/10^A058183(k)). - Stefano Spezia, Nov 30 2022

Extensions

More terms from Sascha Kurz, Jan 04 2003
More terms from Michel Marcus, Jan 15 2019

A078261 a(n) = numerator(N) where N = 0.246...(2n) is the concatenation of the first n even numbers after the decimal point.

Original entry on oeis.org

1, 6, 123, 617, 24681, 6170253, 1234050607, 30851265177, 12340506070809, 123405060708091, 123405060708091011, 1542563258851137639, 1234050607080910111213, 61702530354045505560657, 2468101214161820222426283, 308512651770227527803285379, 123405060708091011121314151617
Offset: 1

Views

Author

Amarnath Murthy, Nov 24 2002

Keywords

Crossrefs

Cf. A078258 (similar, with concatenation of 1 to n), A078260 (denominators).

Programs

  • Maple
    a:= n-> (t-> numer(t/10^length(t)))(parse(cat(2*i$i=1..n))):
    seq(a(n), n=1..17);  # Alois P. Heinz, Jun 25 2025
  • PARI
    a(n) = {my(s = ""); for (k=1, n, s = concat(s, Str(2*k))); numerator(eval(s)/10^(#s));} \\ Michel Marcus, Jan 15 2019

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
More terms from Michel Marcus, Jan 15 2019

A172506 a(n) = numerator of fraction a/b, where gcd(a, b) = 1, whose decimal representation has the form (1)(2)(3)...(n-1)(n).(1)(2)(3)...(n-1)(n).

Original entry on oeis.org

11, 303, 123123, 6170617, 246902469, 1929001929, 12345671234567, 617283906172839, 123456789123456789, 123456789101234567891, 12345678910111234567891011, 15432098637639015432098637639, 1234567891011121312345678910111213, 6172839455055606570617283945505560657
Offset: 1

Views

Author

Jaroslav Krizek, Feb 05 2010

Keywords

Comments

Sequence of denominators: 10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, ... Conjecture: this sequence is not equal to the sequence A078257.
From Michael S. Branicky, Nov 30 2022: (Start)
The conjecture is false: the denominators here are the same as in A078257.
Proof. Let Cn denote the concatenation (1)(2)(3)...(n-1)(n) and en its number of decimal digits. The unreduced numerator and denominator for a(n) are Cn and 10^en, respectively. For A078257(n), they are Cn*(10^en + 1) and 10^en. Since (10^en + 1) is never divisible by 2 or 5, no reductions can be made in the denominator of A078257(n) beyond those allowed by the unreduced numerator of a(n). (End)

Examples

			a(6) = 1929001929; 1929001929/15625 = 123456.123456.
		

Crossrefs

Programs

  • Python
    from itertools import count, islice
    def agen(): # generator of terms
        k, den, pow = 0, 1, 0
        for n in count(1):
            sn = str(n)
            k = k*10**len(sn) + n
            den *= 10**len(sn)
            pow += len(sn)
            nr, c2, c5 = k*(den+1), pow, pow
            while nr%2 == 0 and c2 > 0: nr //= 2; c2 -= 1
            while nr%5 == 0 and c5 > 0: nr //= 5; c5 -= 1
            yield nr
    print(list(islice(agen(), 19))) # Michael S. Branicky, Nov 30 2022

Extensions

a(9) and beyond from Michael S. Branicky, Nov 30 2022
Showing 1-3 of 3 results.