A280052 -id:A280052 - cp's OEIS frontend

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.

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.

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

Original entry on oeis.org

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

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

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

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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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