login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A297560 Numbers with nonzero Conway base-13 function value. 1
1715, 1728, 1741, 1754, 1767, 1780, 1793, 1806, 1819, 1847, 1848, 1849, 1850, 1851, 1852, 1853, 1854, 1855, 1884, 1897, 1910, 1923, 1936, 1949, 1962, 1975, 1988, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023, 2024 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms below 13^k are of two forms: (a) 10*13^k + Sum_{i=0..k-1} d_i*13^i or 11*13^k + Sum_{i=0..k-1} d_i*13^i, where exactly one of d_0, d_1, ..., d_(k-1) is equal to 12, and the other k - 2 digits are from 0 to 9 (but not all 0); (b) d*13^k + a, where 0 <= d <= 12, and a is any term below 13^(k-1). - Jianing Song, Jun 28 2018
Let b(k) be the number of terms below 13^k, then by the comment above, b(k) = 13*b(k-1) + 2*(k-1)*(10^(k-2)-1), where b(1) = 0. So we have b(k) = (15*13^k - (48*k+160)*10^(k-1) + 12*k + 1)/72, which means that there are (5/24)*N + o(N) terms in this sequence that are no greater than N. - Jianing Song, Jun 04 2019
LINKS
Jianing Song, Table of n, a(n) for n = 1..1062 (All terms below 13^4, a(604)-a(693), a(937)-a(1026) corrected by Jianing Song, May 06 2021.)
Lyam K. Boylan, Arithmetic Digit Manipulation and The Conway Base-13 Function, arXiv:2103.10249 [math.GM], 2021.
FORMULA
a(i*b(k)+j) = i*13^n + a(j), where b(k) is the number of terms below 13^n (see comment), 0 <= i <= 9, 1 <= j <= b(k).
EXAMPLE
1715 in decimal equals A1C in base 13 and has Conway base-13 function value 1 by definition, so 1715 is a term.
2024 in decimal equals BC9 in base 13 and has Conway base-13 function value -0.9, so 2024 is a term.
PROG
(PARI) Conway(n) = my(v=digits(n, 13), d=#v, s=0, p=0); forstep(i=d, 1, -1, if(v[i]==10 || v[i]==11, s=i; break())); forstep(i=d, 1, -1, if(v[i]==12, p=i; break())); [s, p]; if(p>s && s && !sum(i=s+1, p-1, v[i]==12), (sum(i=s+1, p-1, v[i]*10^(p-1-i)) + sum(i=p+1, d, v[i]*10^(p-i)))*if(v[s]==10, 1, -1), 0)
isA297560(n) = !(!Conway(n)) \\ Jianing Song, Jun 04 2019
CROSSREFS
Sequence in context: A221481 A008744 A184090 * A157247 A347969 A267201
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Apr 12 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)