A279732 -id:A279732 - cp's OEIS frontend

A278742 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 10.

1, 2, 3, 10, 11, 12, 20, 30, 100, 110, 200, 300, 1000, 1100, 2000, 2100, 3000, 10000, 20000, 30000, 100000, 110000, 120000, 200000, 300000, 1000000, 1100000, 2000000, 3000000, 10000000, 11000000, 20000000, 21000000, 30000000, 100000000, 200000000, 300000000

Offset: 1

Rémy Sigrist, Nov 27 2016

			The first terms, alongside their partial sums, are:
n     a(n)     Partial sums (A280730)
--    -----    ----------------------
1         1              1
2         2              3
3         3              6
4        10             16
5        11             27
6        12             39
7        20             59
8        30             89
9       100            189
10      110            299
11      200            499
12      300            799
13     1000           1799
14     1100           2899
15     2000           4899
16     2100           6999
17     3000           9999
--    -----          -----
18    10000          19999
19    20000          39999
20    30000          69999

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10000).

Cf. A278743, A279732, A280730.

f[a_] := Function[w, Function[s, Total@ Map[PadLeft[#, s] &, w]]@ Max@ Map[Length, w]]@ Map[IntegerDigits, a]; g[n_] := FixedPoint[Function[k, If[Total@ Drop[RotateRight@ DigitCount@ k, 4] > 0, k + (6 * 10^(Position[#, 4][[1, 1]] - 1)) &@ Reverse@ IntegerDigits@ k, k]], n]; a = {1}; Do[If[n <= 17, k = g[Max@ a + 1]; While[Max@ f@ Join[a, {k}] > 9, k = g[k + 1]], k = 10^4 * a[[n - 17]]]; AppendTo[a, k], {n, 2, 37}]; a (* Michael De Vlieger, Dec 18 2016 *)

a(n)=if(n>17, a(n-17)*10000, [1, 2, 3, 10, 11, 12, 20, 30, 100, 110, 200, 300, 1000, 1100, 2000, 2100, 3000][n]) \\ Charles R Greathouse IV, Nov 27 2016

Vec(x*(1 +2*x +3*x^2 +10*x^3 +11*x^4 +12*x^5 +20*x^6 +30*x^7 +100*x^8 +110*x^9 +200*x^10 +300*x^11 +1000*x^12 +1100*x^13 +2000*x^14 +2100*x^15 +3000*x^16) / (1 -10000*x^17) + O(x^50)) \\ Colin Barker, Jan 10 2017

a(n+17) = 10000*a(n) for any n>0.
a(17k+1) = 10^(4k), k >= 0. - N. J. A. Sloane, Jan 06 2017
G.f.: x*(1 +2*x +3*x^2 +10*x^3 +11*x^4 +12*x^5 +20*x^6 +30*x^7 +100*x^8 +110*x^9 +200*x^10 +300*x^11 +1000*x^12 +1100*x^13 +2000*x^14 +2100*x^15 +3000*x^16) / (1 -10000*x^17). - Colin Barker, Jan 10 2017

A278743 For a base n>1: consider the lexicographically least strictly increasing sequence c_n such that, for any m>0, Sum_{k=1..m} c_n(k) can be computed without carries in base n; the sequence c_n is (conjecturally) eventually linear, and a(n) gives its order.

Original entry on oeis.org

1, 3, 2, 5, 9, 3, 7, 4, 17, 4, 9, 15, 21, 11, 5, 11, 25, 25, 13, 7, 6, 13, 7, 29, 15, 16, 25, 7, 15, 9, 33, 17, 10, 28, 57, 8, 17, 49, 37, 19, 10, 31, 63, 21, 9, 19, 43, 41, 21, 34, 34, 69, 23, 12, 10, 21, 13, 45, 23, 51, 37, 75, 25, 13, 67, 11, 23, 42, 49, 25

Offset: 2

Rémy Sigrist, Nov 27 2016

More precisely, we conjecture that, for any n>1, there are two constants k0 and b such that c_n(k + a(n)) = c_n(k)*n^b for any k>k0. [Corrected by Rémy Sigrist, Dec 24 2016]

For the values of k0 and b see A280051 and A280052. - N. J. A. Sloane, Jan 06 2017

			c_2 = A000079, and A000079 has order 1, hence a(2)=1.
c_10 = A278742, and A278742 has order 17, hence a(10)=17.
See also Links section.

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Rémy Sigrist, PERL program for A278743
Rémy Sigrist, Illustration of the initial terms
Rémy Sigrist, Plot of A278743 vs A280051
N. J. A. Sloane, Table of n, a(n), k0(n), b(n) for n=2..42 (A précis of Sigrist's "Illustration" file)

Cf. A000079, A000124, A278742, A279732, A280051, A280052.

a(A000124(n)) = n for any n>0.

a(A000124(n)+1) = 2*n + 1 for any n>0.

A336207 Lexicographically earliest sequence of nonnegative terms such that whenever a(k_1) = ... = a(k_m) with k_1 < ... < k_m, the sum k_1 + ... + k_m can be computed without carries in factorial base.

Original entry on oeis.org

0, 0, 1, 2, 3, 0, 2, 0, 4, 5, 6, 1, 5, 4, 7, 8, 9, 3, 10, 11, 12, 13, 14, 0, 8, 1, 11, 10, 15, 0, 16, 7, 17, 16, 18, 2, 19, 17, 20, 21, 22, 15, 23, 24, 25, 26, 27, 0, 13, 12, 24, 19, 28, 1, 21, 20, 29, 30, 31, 6, 30, 29, 32, 33, 34, 28, 35, 36, 37, 38, 39, 2

Offset: 1

Rémy Sigrist, Jul 12 2020

			In factorial base:
- 1 = "1", 2 = "10", 3 = "11", 4 = "20",
- we can add without carry 1 and 2, so a(1) = a(2) = 0,
- 1 + 2 + 3 implies a carry, so a(3) = 1,
- 1 + 2 + 4 and 3 + 4 imply a carry, so a(4) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation
Rémy Sigrist, C program for A336207

See A279125 and A336206 for similar sequences.

Cf. A279732.

C
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See Links section.
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a(n) = 0 iff n belongs to A279732.

A343522 Lexicographically least strictly increasing sequence such that, for any n > 0, Sum_{k = 1..n} 1/a(k) can be computed without carries in factorial base.

Original entry on oeis.org

1, 2, 3, 7, 45, 631, 399168, 97044480, 55794106368

Offset: 1

Rémy Sigrist, Apr 18 2021

This sequence is infinite as factorial base expansions of rational numbers are terminating.

In decimal base, we would end after four terms: 1, 2, 3, 6.

			The first terms, alongside the factorial base expansion of 1/a(n), are:
   n  a(n)  fact(1/a(n))
   -  ----  ------------------
   1     1  1
   2     2  0.1
   3     3  0.0 2
   4     7  0.0 0 3 2 0 6
   5    45  0.0 0 0 2 4

Rémy Sigrist, PARI program for A343522
Wikipedia, Factorial number system (Fractional values)
Index entries for sequences related to factorial base representation

Cf. A294168, A279732.

PARI
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See Links section.
```

Sum_{k = 1..n} 1/a(n) < 2.

cp's OEIS Frontend

A278742 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 10.

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A278743 For a base n>1: consider the lexicographically least strictly increasing sequence c_n such that, for any m>0, Sum_{k=1..m} c_n(k) can be computed without carries in base n; the sequence c_n is (conjecturally) eventually linear, and a(n) gives its order.

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Examples

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Formula

A336207 Lexicographically earliest sequence of nonnegative terms such that whenever a(k_1) = ... = a(k_m) with k_1 < ... < k_m, the sum k_1 + ... + k_m can be computed without carries in factorial base.

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Author

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Examples

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Programs

C

Formula

A343522 Lexicographically least strictly increasing sequence such that, for any n > 0, Sum_{k = 1..n} 1/a(k) can be computed without carries in factorial base.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula