A094405 -id:A094405 - cp's OEIS frontend

A115042 Positions of 1's in A094405, "a(n) = sum of all previous terms modulo n, a(1)=1".

1, 2, 18, 24, 35, 40, 252

Offset: 1

Zak Seidov, Feb 27 2006

Positions of 0's: 4, 9, 333.

Res:= 1;
s:= 1:
for n from 2 to 397 do
  a:= s mod n;
  s:= s+a;
  if a = 1 then Res:= Res, n fi;
od:
Res; # Robert Israel, Aug 16 2018

Keywords fini,full added by Robert Israel, Aug 16 2018

A215451 a(0)=1, a(n) = (sum of previous terms) mod (a(n-1)+n).

Original entry on oeis.org

1, 1, 2, 4, 0, 3, 2, 4, 5, 8, 12, 19, 30, 5, 1, 1, 13, 21, 15, 11, 3, 17, 22, 20, 0, 20, 10, 28, 54, 0, 2, 4, 14, 23, 33, 0, 12, 28, 52, 45, 35, 48, 88, 61, 42, 36, 35, 70, 16, 1, 8, 41, 3, 21, 0, 5, 18, 23, 43, 17, 1, 41, 65, 111, 149, 25, 1, 53, 29, 63, 98, 102, 154, 5

Offset: 0

Alex Ratushnyak, Aug 11 2012

nonn

Indices of 0's: 4, 24, 29, 35, 54, 267, 5284, 17827, 43631, 120871, 843813, 1854903, 2226536, 4208775, 5525594, ...
Indices of 1's: 0, 1, 14, 15, 49, 60, 66, 116, 331, 6053, 23760, 288502, 496670, 4281666, ...
Indices such that a(n)=n: 1, 2, 22, 315, 1172, 1441, 1846, 2140, 47376, 593870, 16538298, 111824649, 565597433, 791186876, ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A094405, A066910, A215452.

nxt[{n_,t_,a_}]:=Module[{c=Mod[t,a+n+1]},{n+1,t+c,c}]; NestList[nxt,{0,1,1},80][[All,3]] (* Harvey P. Dale, Dec 30 2017 *)

s = a = 1
for n in range(1,333):
    print(a, end=', ')
    a = s % (a+n)
    s += a

a(0)=1, a(n) = (a(0)+...+a(n-1)) mod (a(n-1)+n).

A215452 a(1)=1, a(n) = (sum of previous terms) mod (a(n-1) + n).

Original entry on oeis.org

1, 1, 2, 4, 8, 2, 0, 2, 9, 10, 18, 27, 4, 16, 11, 7, 2, 4, 13, 9, 0, 18, 4, 4, 2, 10, 3, 5, 26, 54, 21, 32, 4, 29, 42, 14, 10, 44, 57, 44, 63, 6, 5, 10, 52, 23, 32, 44, 64, 74, 71, 33, 18, 60, 93, 29, 46, 48, 60, 84, 38, 26, 39, 46, 83, 81, 25, 59, 93, 22, 47, 24, 34, 53

Offset: 1

Alex Ratushnyak, Aug 11 2012

nonn

Indices of 0's: 7, 21, 956, 1576, 1964, 55346, 70460, 99845, 399114, 544095, 35321849, 77073660, ...
Indices of 1's: 1, 2, 129, 193, 262, 4495, 99781, 651856, 35351437, ...
Indices such that a(n)=n: 1, 4, 9, 10, 32, 176, 266, 414, 432, 440, 858, 5953, 6030, 15146, 1408096, 3138130, 35659404, 44722350, 109021513, 790542727, ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A094405, A066910, A215451.

s:= proc(n) option remember; `if`(n<1, 0, s(n-1)+a(n)) end:
a:= proc(n) option remember; `if`(n=1, 1, irem(s(n-1), a(n-1)+n)) end:
seq(a(n), n=1..74);  # Alois P. Heinz, Jun 16 2025

nxt[{t_,n_,a_}]:={t+a,n+1,Mod[t+a,a+n+1]}; NestList[nxt,{0,1,1},80][[All,3]] (* Harvey P. Dale, Sep 01 2016 *)

s = a = 1
for n in range(2,333):
    print(a, end=", ")
    a = s % (a+n)
    s += a

a(1)=1, a(n) = (a(0)+...+a(n-1)) mod (a(n-1)+n).

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).

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A115042 Positions of 1's in A094405, "a(n) = sum of all previous terms modulo n, a(1)=1".

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A215451 a(0)=1, a(n) = (sum of previous terms) mod (a(n-1)+n).

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A215452 a(1)=1, a(n) = (sum of previous terms) mod (a(n-1) + n).

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Maple

Mathematica

Python

Formula

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

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