A008344 -id:A008344 - cp's OEIS frontend

A330725 a(0) = 0; thereafter a(n) = a(n-1) + sigma(n) if sigma(n) > a(n-1), otherwise a(n) = a(n-1) - sigma(n), where sigma is the sum of divisors function A000203.

0, 1, 4, 0, 7, 1, 13, 5, 20, 7, 25, 13, 41, 27, 3, 27, 58, 40, 1, 21, 63, 31, 67, 43, 103, 72, 30, 70, 14, 44, 116, 84, 21, 69, 15, 63, 154, 116, 56, 0, 90, 48, 144, 100, 16, 94, 22, 70, 194, 137, 44, 116, 18, 72, 192, 120, 0, 80, 170, 110, 278, 216, 120, 16

Offset: 0

Alois P. Heinz, Jan 11 2020

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000203, A008344, A008345, A008348, A076042, A111466, A126646, A327442, A331165.

a:= proc(n) option remember; `if`(n=0, 0, ((s, t)-> s+
      `if`(s

nxt[{n_,a_}]:={n+1,If[DivisorSigma[1,n+1]>a,a+DivisorSigma[1,n+1],a- DivisorSigma[ 1,n+1]]}; NestList[nxt,{0,0},70][[All,2]] (* Harvey P. Dale, May 14 2022 *)

A331165 a(n) = a(n-1) + p(n) if p(n) > a(n-1), otherwise a(n) = a(n-1) - p(n), where p is the partition function A000041 (assuming a(n) = 0 for n < 0).

Original entry on oeis.org

1, 0, 2, 5, 0, 7, 18, 3, 25, 55, 13, 69, 146, 45, 180, 4, 235, 532, 147, 637, 10, 802, 1804, 549, 2124, 166, 2602, 5612, 1894, 6459, 855, 7697, 16046, 5903, 18213, 3330, 21307, 42944, 16929, 48114, 10776, 55359, 2185, 65446, 140621, 51487, 157045, 32291, 179564

Offset: 0

Alois P. Heinz, Jan 11 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A008344, A008345, A008348, A076042, A111466, A126646, A327442, A330725.

a:= proc(n) option remember; `if`(n<0, 0, ((s, t)-> s+
     `if`(s

a[n_] := a[n] = If[n<0, 0, With[{a1 = a[n-1], p = PartitionsP[n]}, If[p>a1, a1 + p, a1 - p]]];
a /@ Range[0, 70] (* Jean-François Alcover, Jan 05 2021 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(p = numbpart(n-1)); va[n] = va[n-1] - p; if (va[n] < 0, va[n] += 2*p);); va;} \\ Michel Marcus, Jan 06 2021

A374318 For any n > 0, let b_n(n+1) = 0, and for k = 1..n, if b_n(k+1) >= k then b_n(k) = b_n(k+1) - k otherwise b_n(k) = b_n(k+1) + k; a(n) = b_n(1).

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1

Offset: 0

Rémy Sigrist, Jul 04 2024

nonn

This sequence is a variant of A008344; here we add or subtract by numbers from n down to 1, there by numbers from 1 up to n.

Apparently, the sequence only contains 0's, 1's and 2's.

			The first terms, alongside the corresponding sequences b_n, are:
  n   a(n)  b_n
  --  ----  ----------------------------------
   0     0  [0]
   1     1  [1, 0]
   2     1  [1, 2, 0]
   3     0  [0, 1, 3, 0]
   4     2  [2, 3, 1, 4, 0]
   5     1  [1, 2, 4, 1, 5, 0]
   6     1  [1, 0, 2, 5, 1, 6, 0]
   7     2  [2, 3, 5, 2, 6, 1, 7, 0]
   8     0  [0, 1, 3, 6, 2, 7, 1, 8, 0]
   9     1  [1, 2, 0, 3, 7, 2, 8, 1, 9, 0]
  10     1  [1, 2, 4, 7, 3, 8, 2, 9, 1, 10, 0]

Rémy Sigrist, Colored representation of b_n(k) for n <= 1000 (where the color at (x, y) is function of b_x(y))
Rémy Sigrist, Log-log scatterplot of the ordinal transform of the first 10000 terms

Cf. A008344, A042963, A374317.

a(n) = { my (b = 0); forstep (k = n, 1, -1, if (b >= k, b -= k, b += k);); return (b); }

Empirically, a(n) = 1 iff n belongs to A042963.

A117137 Same as triangle in A117136, but omit final 1 from each row.

Original entry on oeis.org

1, 1, 2, 4, 1, 3, 6, 2, 7, 1, 4, 8, 3, 9, 2, 10, 1, 5, 10, 4, 11, 3, 12, 2, 13, 1, 6, 12, 5, 13, 4, 14, 3, 15, 2, 16, 1, 7, 14, 6, 15, 5, 16, 4, 17, 3, 18, 2, 19, 1

Offset: 0

N. J. A. Sloane, Apr 21 2006

Row n has length 2n+1.

			Triangle begins:
Row 0: 1
Row 1: 1 2 4
Row 2: 1 3 6 2 7
Row 3: 1 4 8 3 9 2 10
Row 4: 1 5 10 4 11 3 12 2 13
...

Cf. A117136, A046901, A008344.

The segments of A046901 appear as rows 2, 7, 22, 67, ... (A060816) of this array.

A329130 a(0)=0; for any n >= 0, if a(n) > n then a(n+1) = a(n) - n, otherwise a(n+1) = a(n) + k, where k is the total number of terms a(m) <= m with m <= n.

Original entry on oeis.org

0, 1, 3, 1, 4, 8, 3, 8, 1, 7, 14, 4, 12, 21, 8, 18, 3, 14, 26, 8, 21, 1, 15, 30, 7, 23, 40, 14, 32, 4, 23, 43, 12, 33, 55, 21, 44, 8, 32, 57, 18, 44, 3, 30, 58, 14, 43, 73, 26, 57, 8, 40, 73, 21, 55, 1, 36, 72, 15, 52, 90

Offset: 0

James Marjamaa, Nov 05 2019

Values where a(n) = n appear to be the values of A027941.

			For n = 0,   a(0) =            0    0 >= a(0)
    n = 1,   a(1) = a(0) + 1 = 1,   1 >= a(1), I = 1
    n = 2,   a(2) = a(1) + 2 = 3,   2 <  a(2), I = 2
    n = 3,   a(3) = a(2) - 2 = 1,   3 >= a(3)
    n = 4,   a(4) = a(3) + 3 = 4,   4 >= a(4), I = 3
    n = 5,   a(5) = a(4) + 4 = 8,   5 <  a(5), I = 4
    n = 6,   a(6) = a(5) - 5 = 3,   6 >= a(6)
    n = 7,   a(7) = a(6) + 5 = 8,   7 <  a(7), I = 5
    n = 8,   a(8) = a(7) - 7 = 1,   8 >= a(8)
    n = 9,   a(9) = a(8) + 6 = 7,   9 >= a(9), I = 6
    n = 10,  a(10)= a(9) + 7 = 14,  10<  a(10), I = 7
    n = 11,  a(11)= a(10)- 10= 4,   11>= a(11)
    n = 12,  a(12)= a(11)+ 8 = 12,  12>= a(12), I = 8
    .
    .
    .

James Marjamaa, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Logarithmic scatterplot of the first 1000000 terms of the ordinal transform of the ordinal transform of the sequence

Cf. A027941, A008344.

#include
void seq(int terms)
{int n = 0; int i = 0; int a = 0; int c = 0;
while (n <= terms)
{
    if (c)
       {int N = n - 1;
        a -= N;
        printf("%d\n", a);}
    else
       {a += i;
        printf("%d\n", a);
        i++;}
    if (a > n)
       {c = 1;}
    else
       {c = 0;}
    n++;
}
}
int main(void)
{
seq(1000);
}

v=k=0; for (n=0, 69, print1 (v, ", "); v=if (v>n, v-n, v+k++)) \\ Rémy Sigrist, Nov 06 2019

a(n+1) = a(n) + I, if n >= a(n) ('I' is the next consecutive integer not yet added);
       = a(n) - n, if n < a(n).
From Yan Sheng Ang, May 20 2020: (Start)
a(A004957(n)) = a(n). Hence it follows that (writing F(n) = A000045(n)):
  a(F(2*n)-1) = F(2*n) for n > 1;
  a(F(2*n)) = 1;
  a(F(2*n+1)-1) = F(2*n+1)-1 (as noted above);
  a(F(2*n+1)) = F(2*n+2);
  a(F(2*n+1)+1) = F(2*n) for n > 1.
(End)

A357839 a(n) is the greatest divisor > 1 of n which has already been listed, otherwise a(n) is the smallest number not yet listed; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 3, 2, 5, 4, 6, 2, 5, 4, 7, 6, 8, 5, 7, 2, 9, 8, 5, 2, 9, 7, 10, 10, 11, 8, 11, 2, 7, 9, 12, 2, 3, 10, 13, 7, 14, 11, 9, 2, 15, 12, 7, 10, 3, 13, 16, 9, 11, 14, 3, 2, 17, 15, 18, 2, 9, 16, 13, 11, 19, 17, 3, 14, 20, 18, 21, 2, 15, 19, 11

Offset: 1

Samuel Harkness, Oct 14 2022

When n is prime, a(n) is the prime index (A000720).

			For n = 6 the set of all divisors of 6 greater than 1 is {2, 3, 6}. Also, the set of all a(n < 6) is {0, 1, 2, 3}. The greatest divisor of 6 (excluding 1) that has been listed is 3, so a(6) = 3.

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, Log-log Scatterplot of the first 3000000 terms
Samuel Harkness, Scatterplot of the first 3000000 terms

Cf. A008336, A008344, A051352.

a = 0; A = {a}; Do[s = Drop[Reverse[Divisors[n]], 1]; s = Drop[s, -1]; If[Length[s] >= 1, Do[If[MemberQ[A, Part[s, d]], AppendTo[A, Part[s, d]]; Break[]], {d, 1, Length[s]}], a++; AppendTo[A, a]], {n, 2, 77}] Print[A]

first(n)=my(v=vector(n),m); forfactored(k=2,n, v[k[1]]=if(vecsum(k[2][,2])==1, m++, my(t); fordiv(k,d, if(d<=m, t=d)); t)); v \\ Charles R Greathouse IV, Oct 14 2022

A363653 a(1) = 1; for n > 1, a(n) = a(n-1) - A000005(n) if a(n) strictly positive, else a(n) = a(n-1) + A000005(n).

Original entry on oeis.org

1, 3, 1, 4, 2, 6, 4, 8, 5, 1, 3, 9, 7, 3, 7, 2, 4, 10, 8, 2, 6, 2, 4, 12, 9, 5, 1, 7, 5, 13, 11, 5, 1, 5, 1, 10, 8, 4, 8, 16, 14, 6, 4, 10, 4, 8, 6, 16, 13, 7, 3, 9, 7, 15, 11, 3, 7, 3, 1, 13, 11, 7, 1, 8, 4, 12, 10, 4, 8, 16, 14, 2, 4, 8, 2, 8, 4, 12, 10, 20, 15, 11, 9, 21

Offset: 1

Ctibor O. Zizka, Jun 13 2023

nonn

Variation on Recamán's sequence A005132.

			a(1) = 1
a(2) = 1 + A000005(2) = 3

Cf. A000005, A005132, A008343, A008344, A046901, A063733, A336760.

a[1] = 1; a[n_] := a[n] = Module[{d = DivisorSigma[0, n]}, If[a[n - 1] > d, a[n - 1] - d, a[n - 1] + d]]; Array[a, 100] (* Amiram Eldar, Jun 13 2023 *)

cp's OEIS Frontend

A330725 a(0) = 0; thereafter a(n) = a(n-1) + sigma(n) if sigma(n) > a(n-1), otherwise a(n) = a(n-1) - sigma(n), where sigma is the sum of divisors function A000203.

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A331165 a(n) = a(n-1) + p(n) if p(n) > a(n-1), otherwise a(n) = a(n-1) - p(n), where p is the partition function A000041 (assuming a(n) = 0 for n < 0).

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A374318 For any n > 0, let b_n(n+1) = 0, and for k = 1..n, if b_n(k+1) >= k then b_n(k) = b_n(k+1) - k otherwise b_n(k) = b_n(k+1) + k; a(n) = b_n(1).

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A117137 Same as triangle in A117136, but omit final 1 from each row.

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A329130 a(0)=0; for any n >= 0, if a(n) > n then a(n+1) = a(n) - n, otherwise a(n+1) = a(n) + k, where k is the total number of terms a(m) <= m with m <= n.

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C

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A357839 a(n) is the greatest divisor > 1 of n which has already been listed, otherwise a(n) is the smallest number not yet listed; a(1) = 0.

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A363653 a(1) = 1; for n > 1, a(n) = a(n-1) - A000005(n) if a(n) strictly positive, else a(n) = a(n-1) + A000005(n).

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