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 1-8 of 8 results.

A102830 "True already", base 4, start 1: a(n) is the least integer such that the sequence up to a(n-1) written in base 4 contains floor(a(n)/4) copies of the digit a(n) % 4, with a(0) = 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 15, 21, 22, 23, 26, 27, 31, 38, 39, 43, 47, 55, 62, 66, 70, 71, 74, 75, 79, 86, 87, 90, 91, 95, 102, 103, 107, 111, 119, 126, 130, 135, 139, 143, 151, 155, 159, 167, 171, 175, 183, 191, 203, 210, 214, 218, 223, 226, 234, 237, 241, 245, 250
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

References

Crossrefs

Cf. A102823-A102829, A102357.

A249626 a(0) = 0, a(n+1) = smallest number, not occurring earlier, containing the smallest of the least frequently occurring digits in all preceding terms.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 34, 35, 36, 37, 38, 39, 40, 45, 46, 47, 48, 49, 50, 56, 57, 58, 59, 60, 67, 68, 69, 70, 78, 79, 80, 89, 90, 100, 21, 31, 41, 51, 61, 71, 81, 91, 22, 32, 42
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 03 2014

Keywords

Comments

a(n) = A102823(n) for n <= 55;
not all numbers occur: all repunits (A002275) greater than 1 are missing; idea of proof: for n > 1 the digit 1 will never again be the smallest of least frequently occurring digits;
A249648 gives positions of terms containing a zero.

Examples

			n = 11: digits 0 and 1 occur twice in {a(k): k=0..10}, all other digits exactly once, where 2 is the smallest; therefore a(11) must contain digit 2, and 12 is the smallest unused number containing 2, hence a(11) = 12.
n = 55: digits 0..9 occur exactly 10 times in {a(k): k=0..54}; therefore a(55) must contain digit 0, the smallest digit; a(55) = 100, as 100 is the smallest unused number containing 0;
n = 56: least occurring digits in {a(k): k=0..10} are 2..9 and 2 is the smallest; therefore a(56) must contain digit 2, and 21 is the smallest unused number containing 2, hence a(56) = 21.

Links

Crossrefs

Cf. A031298, A002275, A011540, A249648, A102823.

Programs

  • Haskell
    import Data.List (delete, group, sortBy); import Data.Function (on)
a249626 n = a249626_list !! n
a249626_list = f (zip [0,0..] [0..9]) a031298_tabf where
   f acds@((,dig):) zss = g zss where
     g (ys:yss) = if dig `elem` ys
                     then y : f acds' (delete ys zss) else g yss
       where y = foldr (\d v -> 10 * v + d) 0 ys
             acds' = sortBy (compare `on` fst) $
                    addd (sortBy (compare `on` snd) acds)
                         (sortBy (compare `on` snd) $
                                 zip (map length gss) (map head gss))
             addd cds [] = cds
             addd []   _ = []
             addd ((c, d) : cds) yys'@((cy, dy) : yys)
                  | d == dy  = (c + cy, d) : addd cds yys
                  | otherwise = (c, d) : addd cds yys'
             gss = sortBy compare $ group ys

A102824 "True already", base 2, start 0: a(n) is the least integer such that the sequence up to a(n-1) written in base 2 contains floor(a(n)/2) copies of the digit a(n) % 2, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 14, 16, 23, 26, 30, 32, 42, 48, 56, 62, 64, 75, 82, 89, 96, 101, 109, 116, 122, 126, 128, 142, 150, 158, 164, 174, 180, 188, 194, 204, 212, 220, 226, 234, 240, 248, 254, 256, 272, 286, 294, 304, 316, 324, 336, 345, 355, 364, 372, 380, 386
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

References

Crossrefs

Cf. A102823-A102830, A102357.

A102827 "True already", base 10, start 1: a(n) is the least integer such that the sequence up to a(n-1) written in base 10 contains floor(a(n)/10) copies of the digit a(n) % 10, with a(0) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128, 129, 133
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

Comments

Conjecture: this sequence in various bases never includes a term divisible by the base.

Examples

			The first 9 values of the sequence written in decimal include no '0's and 1 '1', so the next value cannot be 10 (the count of '0's is not 1) but can be 11.

References

Crossrefs

Cf. A102823-A102830, A102357.

Programs

  • Maple
    A102827aux := proc(n,dig)
    local c,d ;
    c := 0 ;
    for d in convert(n,base,10) do
        if d = dig then
            c := c+1 ;
        end if;
    end do:
    c ;
end proc:
A102827 := proc(n)
    option remember;
    local a,a10,ad,cum;
    if n < 8 then
        return n+1 ;
    end if;
    for a from 1 do
        a10 := floor(a/10) ;
        ad := a mod 10 ;
        cum := add( A102827aux(procname(i),ad),i=0..n-1) ;
        if cum = a10 then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, Mar 30 2014

A102829 "True already", base 3, start 1: a(n) is the least integer such that the sequence up to a(n-1) written in base 3 contains floor(a(n)/3) copies of the digit a(n) % 3, with a(0) = 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 13, 14, 17, 23, 29, 32, 35, 41, 44, 50, 56, 62, 67, 74, 76, 82, 88, 92, 95, 98, 104, 110, 113, 116, 122, 125, 131, 137, 143, 152, 161, 173, 179, 188, 193, 202, 206, 215, 223, 226, 232, 238, 244, 250, 256, 263, 269, 274, 278, 284, 287, 293, 299
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

References

Crossrefs

Cf. A102823-A102830, A102357.

A102825 "True already", base 3, start 0: a(n) is the least integer such that the sequence up to a(n-1) written in base 3 contains floor(a(n)/3) copies of the digit a(n) % 3, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 11, 14, 17, 18, 24, 25, 27, 31, 38, 41, 42, 45, 50, 51, 54, 63, 68, 69, 72, 78, 81, 88, 94, 98, 104, 108, 110, 113, 116, 122, 125, 129, 132, 135, 143, 147, 150, 153, 159, 162, 174, 180, 188, 192, 198, 204, 207, 213, 216, 225, 231
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

References

Crossrefs

Cf. A102823-A102830, A102357.

A102826 "True already", base 4, start 0: a(n) is the least integer such that the sequence up to a(n-1) written in base 4 contains floor(a(n)/4) copies of the digit a(n) % 4, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 16, 18, 19, 22, 23, 26, 27, 31, 32, 39, 40, 43, 44, 48, 55, 56, 60, 64, 69, 70, 74, 75, 79, 86, 87, 90, 91, 95, 96, 103, 104, 107, 108, 112, 119, 120, 124, 128, 138, 139, 143, 151, 152, 155, 156, 160, 163, 167, 171, 172, 176
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

References

Crossrefs

Cf. A102823-A102830, A102357.

A102828 "True already", base 2, start 1: a(n) is the least integer such that the sequence up to a(n-1) written in base 2 contains floor(a(n)/2) copies of the digit a(n) % 2, with a(0) = 1.

Original entry on oeis.org

1, 3, 7, 13, 19, 25, 31, 41, 47, 57, 65, 69, 75, 83, 91, 101, 109, 119, 131, 137, 143, 153, 161, 167, 177, 185, 195, 203, 213, 223, 237, 249, 261, 267, 275, 283, 293, 301, 311, 323, 331, 341, 351, 365, 377, 389, 397, 407, 419, 429, 441, 453, 463, 477, 491
Offset: 0

Views

Author

Hugo van der Sanden, Feb 26 2005

Keywords

References

Crossrefs

Cf. A102823-A102830, A102357.
Showing 1-8 of 8 results.

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