A155043 -id:A155043 - cp's OEIS frontend

A320357 a(0)=1; a(1)=1; for n >= 2, a(n) = a(n-1) + a(n-A000005(n)).

1, 1, 2, 3, 4, 7, 9, 16, 20, 29, 38, 67, 76, 143, 181, 248, 315, 563, 639, 1202, 1383, 1946, 2585, 4531, 4846, 7431, 10016, 14547, 17132, 31679, 34264, 65943, 75959, 107638, 141902, 207845, 222392, 430237, 572139, 779984, 855943, 1635927, 1777829, 3413756, 3985895, 4765879

Offset: 0

Ctibor O. Zizka, Oct 11 2018

nonn

			a(4) = a(3)+a(1) = a(2)+a(1)+a(1) = a(1)+a(0)+a(1)+a(1) = 4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..9000
Vaclav Kotesovec, Graph a(n+1)/a(n)

Cf. A000005, A155043, A320520.

a[n_] := a[n] = If[n < 2, 1, a[n-1] + a[n - DivisorSigma[0, n]]]; Table[a[n], {n, 0, 50}] (* Vaclav Kotesovec, Oct 14 2018 *)

1 <= a(n+1)/a(n) <= 2. - Vaclav Kotesovec, Oct 14 2018

By empirical observation a(n) ~ 3.179662855437*exp(0.3175*n). - Ctibor O. Zizka, Oct 15 2018

A330738 Ordinal transform of A049820, where A049820(n) = n - d(n), with d(n) the number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A049820, A060990, A155043.

b[_] = 0;
a[n_] := a[n] = With[{t = n - DivisorSigma[0, n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 22 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A049820(n) = (n-numdiv(n));
v330738 = ordinal_transform(vector(up_to, n, A049820(n)));
A330738(n) = v330738[n];

A327715 a(0) = 0; for n >= 1, a(n) = 1 + a(n-A009191(n)).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 8, 9, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 19, 20, 20, 21, 19, 20, 20, 20, 21, 22, 22, 23, 23, 24, 24, 25, 20, 21, 21, 21, 22, 23, 23, 24, 25, 26

Offset: 0

Ctibor O. Zizka, Sep 23 2019

nonn

Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k with k - gcd(k,d(k)), where d(k) is the number of divisors of k (A000005).

Empirically: n/log(n) <= a(n) <= n/log(n) + 2*log(n).

			a(6) = 1 + a(6-gcd(6,4)) = 1 + a(4) = 2 + a(4-gcd(4,3)) = 2 + a(3) = 3 + a(3-gcd(3,2)) = 3 + a(2) = 4 + a(2-gcd(2,2)) = 4 + a(0) = 4.

Cf. A000005, A009191, A155043.

a(n) = if (n==0, 0, 1 + a(n - gcd(n, numdiv(n)))); \\ Michel Marcus, Sep 25 2019

A330877 Number of steps needed to reach zero or a cycle when starting from k = n and repeatedly applying the map that replaces k by k - d(k) if k is even, by k + d(k) if k is odd, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 2, 1, 7, 3, 6, 2, 5, 4, 4, 3, 12, 3, 11, 4, 10, 13, 10, 4, 9, 5, 8, 5, 8, 14, 7, 6, 32, 6, 32, 6, 31, 7, 30, 7, 29, 33, 29, 8, 28, 8, 28, 8, 27, 9, 26, 9, 12, 9, 11, 10, 25, 10, 25, 10, 24, 10, 23, 11, 23, 10, 22, 12, 21, 24, 21, 12, 21, 13

Offset: 0

Ctibor O. Zizka, Apr 29 2020

nonn

First cycle we see for n = 83. The length of the cycle is 38 steps. To reach a cycle means the time to first step into the loop.

			n = 1, mapping steps are 1 + 1 = 2, 2 - 2 = 0, a(1) = 2;
n = 2, mapping steps are 2 - 2 = 0, a(2) = 1;
n = 3, mapping steps are 3 + 2 = 5, 5 + 2 = 7, 7 + 2 = 9, 9 + 3 = 12, 12 - 6 = 6, 6 - 4 = 2, 2 - 2 = 0, a(3) = 7;
n = 4, mapping steps are 4 - 3 = 1, 1 + 1 = 2, 2 - 2 = 0, a(4) = 3;
n = 5, mapping steps are 5 + 2 = 7, 7 + 2 = 9, 9 + 3 = 12, 12 - 6 = 6, 6 - 4 = 2, 2 - 2 = 0, a(5) = 6.

Cf. A000005, A155043, A261085, A261088.

Cf. A049820, A062249.

cp's OEIS Frontend

A320357 a(0)=1; a(1)=1; for n >= 2, a(n) = a(n-1) + a(n-A000005(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A330738 Ordinal transform of A049820, where A049820(n) = n - d(n), with d(n) the number of divisors of n (A000005).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A327715 a(0) = 0; for n >= 1, a(n) = 1 + a(n-A009191(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A330877 Number of steps needed to reach zero or a cycle when starting from k = n and repeatedly applying the map that replaces k by k - d(k) if k is even, by k + d(k) if k is odd, where d(k) is the number of divisors of k (A000005).

Views

Author

Keywords

Comments

Examples

Crossrefs