cp's OEIS Frontend

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).

%I A172506 #21 Jul 25 2024 04:32:50
%S A172506 11,303,123123,6170617,246902469,1929001929,12345671234567,
%T A172506 617283906172839,123456789123456789,123456789101234567891,
%U A172506 12345678910111234567891011,15432098637639015432098637639,1234567891011121312345678910111213,6172839455055606570617283945505560657
%N 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).
%C A172506 Sequence of denominators: 10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, ... Conjecture: this sequence is not equal to the sequence A078257.
%C A172506 From _Michael S. Branicky_, Nov 30 2022: (Start)
%C A172506 The conjecture is false: the denominators here are the same as in A078257.
%C A172506 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)
%e A172506 a(6) = 1929001929; 1929001929/15625 = 123456.123456.
%o A172506 (Python)
%o A172506 from itertools import count, islice
%o A172506 def agen(): # generator of terms
%o A172506     k, den, pow = 0, 1, 0
%o A172506     for n in count(1):
%o A172506         sn = str(n)
%o A172506         k = k*10**len(sn) + n
%o A172506         den *= 10**len(sn)
%o A172506         pow += len(sn)
%o A172506         nr, c2, c5 = k*(den+1), pow, pow
%o A172506         while nr%2 == 0 and c2 > 0: nr //= 2; c2 -= 1
%o A172506         while nr%5 == 0 and c5 > 0: nr //= 5; c5 -= 1
%o A172506         yield nr
%o A172506 print(list(islice(agen(), 19))) # _Michael S. Branicky_, Nov 30 2022
%Y A172506 Cf. A078257, A078258.
%K A172506 nonn,base
%O A172506 1,1
%A A172506 _Jaroslav Krizek_, Feb 05 2010
%E A172506 a(9) and beyond from _Michael S. Branicky_, Nov 30 2022