A278742 -id:A278742 - cp's OEIS frontend

A280730 Partial sums of A278742.

1, 3, 6, 16, 27, 39, 59, 89, 189, 299, 499, 799, 1799, 2899, 4899, 6999, 9999, 19999, 39999, 69999, 169999, 279999, 399999, 599999, 899999, 1899999, 2999999, 4999999, 7999999, 17999999, 28999999, 48999999, 69999999, 99999999, 199999999, 399999999, 699999999

Offset: 1

N. J. A. Sloane, Jan 09 2017

Cf. A278742.

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.

A279732 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in factorial base.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 48, 120, 240, 720, 840, 1440, 1560, 5040, 10080, 15120, 40320, 45360, 80640, 120960, 362880, 403200, 725760, 1088640, 3628800, 3991680, 7257600, 7620480, 10886400, 39916800, 43545600, 79833600, 119750400, 159667200, 479001600, 958003200

Offset: 1

Rémy Sigrist, Dec 18 2016

This sequence is to factorial base what A278742 is to base 10.
This sequence contains the factorial numbers (A000142); the corresponding indices are 1, 2, 3, 5, 8, 10, 14, 17, 21, 25, 30, 35, 39, 45, 49, 56, 62, 67, 74, 79, 87, 93, 102, 108, 116, 122, 131, 138, 148, 155, ...
Occasionally, the sum of the first n terms equals A033312(k) for some k;
- In that case: a(n+1)=k!, and k! divides a(m) for any m>n,
- The corresponding indices are 1, 7, 13, 34, 44, 61, 73, 101, 115, 147, 343, 387, 487, 605, 657, 788, 1226, 1296, 1575, 2986, 3586, 5152, 5260, 8236, 9173, ...
- Conjecture: this happens infinitely often.

			The first terms in base 10 and factorial base, alongside their partial sums in factorial base, are:
n    a(n)        a(n) in fact. base      Partial sum in fact. base
--   ---------   ---------------------   -------------------------
1            1                       1                         1
2            2                     1,0                       1,1
3            6                   1,0,0                     1,1,1
4            8                   1,1,0                     2,2,1
5           24                 1,0,0,0                   1,2,2,1
6           30                 1,1,0,0                   2,3,2,1
7           48                 2,0,0,0                   4,3,2,1
8          120               1,0,0,0,0                 1,4,3,2,1
9          240               2,0,0,0,0                 3,4,3,2,1
10         720             1,0,0,0,0,0               1,3,4,3,2,1
11         840             1,1,0,0,0,0               2,4,4,3,2,1
12        1440             2,0,0,0,0,0               4,4,4,3,2,1
13        1560             2,1,0,0,0,0               6,5,4,3,2,1
14        5040           1,0,0,0,0,0,0             1,6,5,4,3,2,1
15       10080           2,0,0,0,0,0,0             3,6,5,4,3,2,1
16       15120           3,0,0,0,0,0,0             6,6,5,4,3,2,1
17       40320         1,0,0,0,0,0,0,0           1,6,6,5,4,3,2,1
18       45360         1,1,0,0,0,0,0,0           2,7,6,5,4,3,2,1
19       80640         2,0,0,0,0,0,0,0           4,7,6,5,4,3,2,1
20      120960         3,0,0,0,0,0,0,0           7,7,6,5,4,3,2,1
21      362880       1,0,0,0,0,0,0,0,0         1,7,7,6,5,4,3,2,1
22      403200       1,1,0,0,0,0,0,0,0         2,8,7,6,5,4,3,2,1
23      725760       2,0,0,0,0,0,0,0,0         4,8,7,6,5,4,3,2,1
24     1088640       3,0,0,0,0,0,0,0,0         7,8,7,6,5,4,3,2,1
25     3628800     1,0,0,0,0,0,0,0,0,0       1,7,8,7,6,5,4,3,2,1
26     3991680     1,1,0,0,0,0,0,0,0,0       2,8,8,7,6,5,4,3,2,1
27     7257600     2,0,0,0,0,0,0,0,0,0       4,8,8,7,6,5,4,3,2,1
28     7620480     2,1,0,0,0,0,0,0,0,0       6,9,8,7,6,5,4,3,2,1
29    10886400     3,0,0,0,0,0,0,0,0,0       9,9,8,7,6,5,4,3,2,1
30    39916800   1,0,0,0,0,0,0,0,0,0,0     1,9,9,8,7,6,5,4,3,2,1
31    43545600   1,1,0,0,0,0,0,0,0,0,0    2,10,9,8,7,6,5,4,3,2,1
32    79833600   2,0,0,0,0,0,0,0,0,0,0    4,10,9,8,7,6,5,4,3,2,1
33   119750400   3,0,0,0,0,0,0,0,0,0,0    7,10,9,8,7,6,5,4,3,2,1
34   159667200   4,0,0,0,0,0,0,0,0,0,0   11,10,9,8,7,6,5,4,3,2,1

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A279732
Index entries for sequences related to factorial base representation

Cf. A000142, A007623, A033312, A278742, A278743.

r = MixedRadix[Reverse@ Range[2, 30]]; f[a_] := Function[w, Function[s, Total@ Map[PadLeft[#, s] &, w]]@ Max@ Map[Length, w]]@ Map[IntegerDigits[#, r] &, a]; g[w_] := Times @@ Boole@ MapIndexed[#1 <= First@ #2 &, Reverse@ w] > 0; a = {1}; Do[k = Max@ a + 1; While[! g@ f@ Join[a, {k}], k++]; AppendTo[a, k], {n, 2, 16}]; a (* Michael De Vlieger, Dec 18 2016 *)

A336206 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 base 10.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 0, 0, 0, 2, 6, 6, 7, 8, 9, 10, 0, 3, 7, 11, 11, 12, 13, 14, 15, 16, 0, 3, 8, 12, 17, 17, 18, 19, 20, 21, 1, 4, 11, 13, 22, 22, 23, 24, 25, 26, 1, 7, 14, 18, 27, 28, 29, 30, 31, 32, 2, 9, 19, 33, 34, 35, 36, 37, 38, 39, 5, 15, 40, 41

Offset: 1

Rémy Sigrist, Jul 12 2020

This sequence is a decimal variant of A279125.

			We can choose a(1) = a(2) = a(3) = 0 as 1 + 2 + 3 = 6 can be computed without carries.
However 1 + 2 + 3 + 4 implies a carry, so a(4) = 1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 100000 terms
Rémy Sigrist, C program for A336206

Cf. A278742, A279125.

C
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See Links section.
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a(n) = 0 iff n belongs to A278742.

A280731 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 9 (the numbers are written in base 10).

Original entry on oeis.org

1, 2, 3, 9, 10, 18, 19, 81, 90, 162, 171, 729, 810, 1458, 1539, 6561, 7290, 13122, 13851, 59049, 65610, 118098, 124659, 531441, 590490, 1062882, 1121931, 4782969, 5314410, 9565938, 10097379, 43046721, 47829690, 86093442, 90876411, 387420489, 430467210

Offset: 1

N. J. A. Sloane, Jan 09 2017

Base 9 analog of A278742.

Colin Barker, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (0,0,0,9).

Cf. A278742, A278743, A280051, A280052.

See A281366 for these numbers written in base 9.

LinearRecurrence[{0,0,0,9},{1,2,3,9,10,18,19},50] (* Harvey P. Dale, Feb 28 2022 *)

Vec(x*(1 + 2*x + 3*x^2 + 9*x^3 + x^4 - 8*x^6) / ((1 - 3*x^2)*(1 + 3*x^2)) + O(x^50)) \\ Colin Barker, Jan 10 2017

For k>7, a(k+4) = 9*a(k).

G.f.: x*(1 + 2*x + 3*x^2 + 9*x^3 + x^4 - 8*x^6) / ((1 - 3*x^2)*(1 + 3*x^2)). - Colin Barker, Jan 10 2017

A281366 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 9 (the numbers are written in base 9).

Original entry on oeis.org

1, 2, 3, 10, 11, 20, 21, 100, 110, 200, 210, 1000, 1100, 2000, 2100, 10000, 11000, 20000, 21000, 100000, 110000, 200000, 210000, 1000000, 1100000, 2000000, 2100000, 10000000, 11000000, 20000000, 21000000, 100000000, 110000000, 200000000, 210000000

Offset: 1

N. J. A. Sloane, Jan 28 2017

Base 9 analog of A278742.

Colin Barker, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (0,0,0,10).

Cf. A278742, A278743, A280051, A280052.

See A280731 for these numbers written in base 10.

Vec(x*(1 + 2*x + 3*x^2 + 10*x^3 + x^4 - 9*x^6) / (1 - 10*x^4) + O(x^60)) \\ Colin Barker, Jan 29 2017

From Colin Barker, Jan 29 2017: (Start)
G.f.: x*(1 + 2*x + 3*x^2 + 10*x^3 + x^4 - 9*x^6)/(1 - 10*x^4).
a(n) = 10*a(n-4) for n>7. (End)

More terms from Colin Barker, Jan 29 2017

A368316 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, a(n) and Sum_{k = 0..n-1} a(k) can be added without carries in balanced ternary.

Original entry on oeis.org

0, 1, 2, 5, 3, 15, 4, 6, 41, 9, 18, 10, 125, 12, 16, 8, 45, 13, 14, 369, 27, 54, 28, 11, 7, 126, 17, 55, 26, 1107, 30, 51, 31, 131, 36, 46, 29, 375, 37, 44, 39, 123, 40, 42, 35, 3285, 57, 24, 135, 81, 405, 82, 38, 19, 132, 53, 1134, 84, 25, 134, 85, 23, 378

Offset: 0

Rémy Sigrist, Dec 21 2023

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.
If we restrict ourselves to positive integers and allow duplicates, then we obtain A236313.
This sequence can be seen as a variant of A278742; however, the present sequence is not strictly increasing.
Will every nonnegative integer appear in the sequence?

			The first terms, alongside the balanced ternary expansions of a(n) and b(n) = Sum_{k = 0..n-1} a(k), are:
  n           |  0  1   2    3    4     5     6     7      8      9     10
  a(n)        |  0  1   2    5    3    15     4     6     41      9     18
  bter(b(n))  |  0  0   1   10  10T   11T  100T  1010   1100  100TT  101TT
  bter(a(n))  |  0  1  1T  1TT   10  1TT0    11   1T0  1TTTT    100   1T00

Rémy Sigrist, PARI program
Wikipedia, Balanced ternary

Cf. A236313, A278742.

PARI
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See Links section.
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A345369 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the product of the first n terms can be computed without carry in base 10.

Original entry on oeis.org

1, 2, 3, 10, 11, 100, 101, 1000, 10000, 10001, 100000, 1000000, 10000000, 100000000, 100000001, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 10000000000000001, 100000000000000000

Offset: 1

Rémy Sigrist, Jun 16 2021

This sequence is a variant of A278742; here we multiply, there we add.

This sequence is the union of {2, 3}, A080176 and A011557.

			The first terms, alongside their product, are:
  n   a(n)   a(1) * ... * a(n)
  --  -----  ------------------
   1      1                   1
   2      2                   2
   3      3                   6
   4     10                  60
   5     11                 660
   6    100               66000
   7    101             6666000
   8   1000          6666000000
   9  10000      66660000000000
  10  10001  666666660000000000

Cf. A080176, A011557.

cp's OEIS Frontend

A280730 Partial sums of A278742.

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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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A279732 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in factorial base.

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A336206 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 base 10.

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A280731 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 9 (the numbers are written in base 10).

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A281366 Lexicographically least strictly increasing sequence such that, for any n>0, Sum_{k=1..n} a(k) can be computed without carries in base 9 (the numbers are written in base 9).

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A368316 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, a(n) and Sum_{k = 0..n-1} a(k) can be added without carries in balanced ternary.

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A345369 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the product of the first n terms can be computed without carry in base 10.

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