A278743 -id:A278743 - cp's OEIS frontend

A280051 Let c_n(k) be the sequence defined in A278743; here we give the associated values of k0.

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 22, 0, 0, 15, 11, 0, 0, 11, 0, 0, 7, 12, 0, 0, 0, 0, 0, 0, 24, 13, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 27, 0, 0, 29, 0, 0, 0, 15, 0, 0, 29, 29, 0, 0, 15, 0, 0, 18, 16, 0, 0, 31, 31, 34, 0, 0, 16

Offset: 2

N. J. A. Sloane, Jan 06 2017

nonn

Let c_n(k) be the sequence defined in A278743, for n >= 2. It is conjectured that there are numbers k0 and b such that c_n(k) satisfies the recurrence c_n(k + A278743(n)) = c_n(k)*n^b for k > k0. Here we give the values of k0. The values of b are given in A280052.

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Rémy Sigrist, Illustration of the first terms of A278743, A280051, A280052
Rémy Sigrist, Plot of A278743 vs A280051
N. J. A. Sloane, Table of n, A278743(n), k0(n), b(n) for n=2..42 (A précis of Sigrist's "Illustration" file)

Cf. A278743, A280052.

More terms from Rémy Sigrist, Jan 07 2017

A280052 Let c_n(k) be the sequence defined in A278743; here we give the associated values of b.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 1, 4, 1, 2, 3, 4, 2, 1, 2, 4, 4, 2, 1, 1, 2, 1, 4, 2, 2, 3, 1, 2, 1, 4, 2, 1, 3, 6, 1, 2, 5, 4, 2, 1, 3, 6, 2, 1, 2, 4, 4, 2, 3, 3, 6, 2, 1, 1, 2, 1, 4, 2, 4, 3, 6, 2, 1, 5, 1, 2, 3, 4, 2, 1, 3, 6, 2, 1, 5, 6, 1, 2, 3, 4, 2, 2, 3, 6, 2, 1

Offset: 2

N. J. A. Sloane, Jan 06 2017

nonn

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Rémy Sigrist, Illustration of the first terms of A278743, A280051, A280052
N. J. A. Sloane, Table of n, A278743(n), k0(n), b(n) for n=2..42 (A précis of Sigrist's "Illustration" file)

Cf. A278743, A280051.

More terms from Rémy Sigrist, Jan 07 2017

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.

Original entry on oeis.org

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

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 *)

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

cp's OEIS Frontend

A280051 Let c_n(k) be the sequence defined in A278743; here we give the associated values of k0.

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A280052 Let c_n(k) be the sequence defined in A278743; here we give the associated values of b.

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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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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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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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