A117846 -id:A117846 - cp's OEIS frontend

A253387 A "mod sequence" where a(n) is the eventual constant value attained by the sequence defined as b(1) = n, b(m) = (sum_{k=1..m-1} b(k)) mod m, with a(n) = -1 in case a constant run is not found.

97, 97, 1, 1, 2, 2, 2, 2, 316, 316, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 12, 12, 4, 4, 4, 4, 12, 12, 11, 11, 11, 11, 316, 316, 11, 11, 316, 316, 316, 316, 6, 6, 316, 316, 316, 316, 316, 316, 316, 316, 97, 97, 316, 316, 316, 316, 13, 13, 316, 316, 13

Offset: 1

Jean-François Alcover, Dec 31 2014

sign

			a(5) = 2, because the b sequence is 5, 1, 0, 2, 3, 5, 2, 2, 2, 2, 2, ...

Jean-François Alcover, Table of n, a(n) for n = 1..1000
MathOverflow, Mod sequences that seem to become constant

Cf. A074482, A117846.

Clear[a]; constantLength = 10; kMax = 2000; a[n_] := a[n] = Module[{k}, Clear[b]; For[ b[1] = n; b[m_] := b[m] = Mod[Sum[b[j], {j, 1, m-1}], m]; k = constantLength, k <= kMax, k++, If[Equal @@ Table[b[k-j], {j, 0, constantLength-1}], Print["a(", n, ") = ", b[k], ", k = ", k - constantLength+1]; Return[b[k]]]]; Print["a(", n, ") = ", -1, ", k = ", k - constantLength+1, " constant run not found"]; Return[-1]]; Table[a[n], {n, 1, 100}]

A372865 a(n) = (Sum_{k = 1..n-1} k*a(k)) (mod n) with a(1) = 1.

Original entry on oeis.org

1, 1, 0, 3, 0, 3, 5, 4, 1, 9, 1, 6, 9, 7, 2, 15, 2, 9, 13, 10, 3, 21, 3, 12, 17, 13, 4, 27, 4, 15, 21, 16, 5, 33, 5, 18, 25, 19, 6, 39, 6, 21, 29, 22, 7, 45, 7, 24, 33, 25, 8, 51, 8, 27, 37, 28, 9, 57, 9, 30, 41, 31, 10, 63, 10, 33, 45, 34, 11, 69, 11, 36, 49, 37, 12, 75, 12, 39, 53, 40

Offset: 1

Peter Bala, May 15 2024

Compare with A066910.
More generally, define a sequence {a(n, s) : n >= 1} with starting parameter s by a(n, s) = (Sum_{k = 1..n-1} k*a(k, s)) (mod n) with a(1, s) = s. The sequence {a(n, s)} is conjectured to be one of 3 types as illustrated by the following examples for s in [1..100].
1) It is easy to verify that the sequence {a(n, 8)} = {8, 0, 2, 2, 2, 2, ...} becomes constant at n = 3 and the sequence {a(n, 38)} = {38, 0, 2, 0, 4, 4, 4, ...} becomes constant at n = 5.
2) For s in {2, 5, 20, 21, 22, 31, 33, 34, 35, 36, 40, 42, 60, 65, 85, 87, 88, 92, 93, 97, 98, 100} the sequence {a(n, s)} appears to be quasipolynomial in n with 6 constituent polynomials of degree 1.
3) For the remaining values of s <= 100, the sequence {a(n, s)} appears to be an eventually periodic sequence with period 6, so again quasipolynomial in n with 6 constituent polynomials of degree 0. For example, an easy induction argument shows that {a(n, 3)} = {3, 1, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 1, ...} has period 6 starting at n = 2.

MathOverflow, Mod sequences that seem to become constant

Cf. A066910, A073117, A094405, A074482, A117846, A253387.

a := proc(n, s) option remember; if n = 1 then s else irem(add(k*a(k, s), k = 1 .. n-1), n) end if; end proc:
seq(a(n, 1), n = 1..80);

CoefficientList[Series[x(x^8 + 4*x^7 + 4*x^6 + 2*x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/((x + 1)^2*(x - 1)^2*(x^2 - x + 1)*(x^2 + x + 1)^2),{x,0,80}],x] (* Stefano Spezia, May 18 2024 *)

lista(nn) = my(v=vector(nn)); v[1]=1; for(n=2, nn, v[n]=sum(k=1, n-1, k*v[k])%n); v; \\ Michel Marcus, May 18 2024

a(n) is quasipolynomial in n (proved by induction): a(6*n) = 3*n for n >= 1, and for n >= 0, a(6*n+1) = 4*n + 1, a(6*n+2) = 3*n + 1, a(6*n+3) = n, a(6*n+4) = 6*n + 3 and a(6*n+5) = n.

G.f.: A(x) = x*(x^8 + 4*x^7 + 4*x^6 + 2*x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/((x + 1)^2*(x - 1)^2*(x^2 - x + 1)*(x^2 + x + 1)^2).

cp's OEIS Frontend

A253387 A "mod sequence" where a(n) is the eventual constant value attained by the sequence defined as b(1) = n, b(m) = (sum_{k=1..m-1} b(k)) mod m, with a(n) = -1 in case a constant run is not found.

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A372865 a(n) = (Sum_{k = 1..n-1} k*a(k)) (mod n) with a(1) = 1.

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