A316156 -id:A316156 - cp's OEIS frontend

A318872 Partial sums of A316156.

1, 3, 6, 12, 19, 27, 36, 49, 64, 81, 99, 118, 138, 169, 201, 240, 280, 323, 378, 437, 500, 564, 629, 695, 762, 830, 899, 969, 1040, 1112, 1185, 1271, 1376, 1482, 1589, 1697, 1806, 1916, 2027, 2145, 2280, 2416, 2553, 2709, 2866, 3024, 3183, 3343, 3504, 3666, 3829, 3993, 4158, 4324, 4491, 4675, 4860, 5046, 5233, 5428

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A316156.

up_to = 10000;
povisin(v,n) = { forstep(j=n,1,-1, if(v[j] == n, return(j))); (0); }; \\ Here: povisin = position_of_n_in_strictly_increasing_v
A318872list(up_to) = { my(v316156 = vector(up_to), v318872 = vector(up_to), k, s); v316156[1] = v318872[1] = 1; for(n=2, up_to, k = 1+v316156[n-1]; if(povisin(v316156, n-1), s = v318872[n-1]; while((s+k)%(n-1), k++)); v316156[n] = k; v318872[n] = v318872[n-1] + v316156[n]); (v318872); };
v318872 = A318872list(up_to);
A318872(n) = v318872[n];

a(n) = Sum_{k=1..n} A316156(k).

A318873 First differences of A316156.

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 11, 1, 7, 1, 3, 12, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 19, 1, 1, 1, 1, 1, 1, 7, 17, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 8, 1, 1, 1, 20, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A316156.

up_to = 20000;
povisin(v,n) = { forstep(j=n,1,-1, if(v[j] == n, return(j))); (0); }; \\ Here: povisin = position_of_n_in_strictly_increasing_v
A318873list(up_to) = { my(v316156 = vector(1+up_to), v318872 = vector(1+up_to), v318873 = vector(up_to), k, s); v316156[1] = v318872[1] = 1; for(n=2, up_to+1, k = 1+v316156[n-1]; if(povisin(v316156, n-1), s = v318872[n-1]; while((s+k)%(n-1), k++)); v316156[n] = k; v318872[n] = v318872[n-1] + v316156[n]; v318873[n-1] = v316156[n] - v316156[n-1]); (v318873); };
v318873 = A318873list(up_to);
A318873(n) = v318873[n];

a(n) = A316156(n+1) - A316156(n).

A316571 a(1) = 1; for n > 1: a(n) = smallest number such that (Sum_{k=1..n} a(k)) is divisible by n - 1.

Original entry on oeis.org

1, 2, 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124

Offset: 1

Jaroslav Krizek, Aug 20 2018

nonn

This is the lexicographically earliest increasing sequence such that n-1 divides the sum of the first n terms.
Sequence b(n) of the sums of the first n+1 terms, Sum_{k=1..n+1} a(k): 3, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, ...
Sequence c(n) of quotients when a(n) is calculated, (Sum_{k=1..n+1} a(k) ) / n is 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

			a(1) = 1 because 1 divides the sum of the first 2 (i.e., 1 + 1) terms (a(1) + a(2)) for whatever term a(2) > a(1).
a(2) = 2 because 2 is the smallest number > a(1) and 2 divides the sum of the first 3 (i.e., 2 + 1) terms (a(1) + a(2) + a(3)) for whatever term a(3) > a(2) such that 2 divides the sum a(1) + a(2) + a(3); the smallest number > a(2) with this property for a(3) is 3.
a(3) = 3.
a(4) = 6 because 6 is the smallest number > a(3) such that 3 divides the sum of the first 4 (i.e., 3 + 1) terms.
a(5) = 8 because 8 is the smallest number > a(4) such that 4 divides the sum of the first 5 (i.e., 4 + 1) terms.

Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A005843, A020739, A316156.

CoefficientList[Series[(1 + 2 x^3 - x^4)/(-1 + x)^2 , {x, 0, 50}], x] (* Stefano Spezia, Sep 16 2018 *)
Join[{1, 2, 3}, LinearRecurrence[{2, -1}, {6, 8}, 60]] (* Jean-François Alcover, Dec 30 2018 *)

a(1)=1, a(2)=2, a(3)=3, a(n)=2(n-1) for n >= 4.
a(n) = A005843(n-1) for n >= 4. - Antti Karttunen, Sep 16 2018
G.f.: (1 + 2*x^3 - x^4)/(-1 + x)^2. - Stefano Spezia, Sep 16 2018
a(n) = A020739(n-4) = 2*(n - 4) + 6 for n >= 4. - Georg Fischer, Jan 19 2019

Definition clarified by Georg Fischer and Alois P. Heinz, Jan 19 2019

cp's OEIS Frontend

A318872 Partial sums of A316156.

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A318873 First differences of A316156.

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A316571 a(1) = 1; for n > 1: a(n) = smallest number such that (Sum_{k=1..n} a(k)) is divisible by n - 1.

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