A329662 -id:A329662 - cp's OEIS frontend

A331410 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k + k/p, where p is the largest prime factor of k.

0, 0, 1, 0, 2, 1, 1, 0, 2, 2, 2, 1, 2, 1, 3, 0, 3, 2, 3, 2, 2, 2, 2, 1, 4, 2, 3, 1, 4, 3, 1, 0, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 2, 1, 2, 4, 4, 2, 4, 3, 4, 1, 4, 4, 4, 3, 2, 1, 3, 0, 4, 3, 4, 3, 3, 3, 3, 2, 5, 4, 5, 3, 3, 3, 3, 2, 4, 3, 3, 2, 5, 3, 5, 2, 5, 4, 3, 2, 2, 2, 5, 1, 3, 2, 4, 4, 5, 4, 3, 2, 4

Offset: 1

Ali Sada, Jan 16 2020

nonn

Let f(n) = A000265(n) be the odd part of n. Let p be the largest prime factor of k, and say k = p * m. Suppose that k is not a power of 2, i.e., p > 2, then f(k) = p * f(m). The iteration is k -> k + k/p = p*m + m = (p+1) * m. So, p * f(m) -> f(p+1) * f(m). Since for p > 2, f(p+1) < p, the odd part in each iteration decreases, until it becomes 1, i.e., until we reach a power of 2. - Amiram Eldar, Feb 19 2020
Any odd prime factor of k can be used at any step of the iteration, and the result will be same. Thus, like A329697, this is also fully additive sequence. - Antti Karttunen, Apr 29 2020
If and only if a(n) is equal to A005087(n), then sigma(2n) - sigma(n) is a power of 2. (See A336923, A046528). - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is [15,18,24,32], taking 3 iterations to reach 32. So, a(15) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree of a(n) labeling the index n and applying a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value of a(n) for n = 1..16383.

Cf. A000265, A005087, A006530 (greatest prime factor), A052126, A078701, A087436, A329662 (positions of records and the first occurrences of each n), A334097, A334098, A334108, A334861, A336467, A336921, A336922, A336923 (A046528).
Cf. array A335430, and its rows A335431, A335882, and also A335874.
Cf. also A329697 (analogous sequence when using the map k -> k - k/p), A335878.
Cf. also A330437, A335884, A335885, A336362, A336363 for other similar iterations.

f:=func; g:=func; a:=[]; for n in [1..1000] do k:=n; s:=0; while not g(k) do  s:=s+1; k:=f(k); end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 19 2020

a[n_] := -1 + Length @ NestWhileList[# + #/FactorInteger[#][[-1, 1]] &, n, # / 2^IntegerExponent[#, 2] != 1 &]; Array[a, 100] (* Amiram Eldar, Jan 16 2020 *)

A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 29 2020

A331410(n) = { my(k=0); while(bitand(n,n-1), k++; my(f=factor(n)[, 1]); n += (n/f[2-(n%2)])); (k); }; \\ Antti Karttunen, Apr 29 2020

A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(1+f[k,1])))); }; \\ Antti Karttunen, Apr 30 2020

From Antti Karttunen, Apr 29 2020: (Start)
This is a completely additive sequence: a(2) = 0, a(p) = 1+a(p+1) for odd primes p, a(m*n) = a(m)+a(n), if m,n > 1.
a(2n) = a(A000265(n)) = a(n).
If A209229(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(n+A052126(n)), or equally, 1 + a(n+(n/A078701(n))).
a(n) = A334097(n) - A334098(n).
a(A122111(n)) = A334108(n).
(End)
a(n) = A334861(n) - A329697(n). - Antti Karttunen, May 14 2020
a(n) = a(A336467(n)) + A087436(n) = A336921(n) + A087436(n). - Antti Karttunen, Mar 16 2021

Data section extended up to a(105) by Antti Karttunen, Apr 29 2020

A335430 Square array where row n lists all numbers k for which A331410(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 9, 15, 16, 12, 10, 17, 25, 32, 14, 11, 19, 29, 73, 64, 24, 13, 27, 37, 75, 125, 128, 28, 18, 30, 45, 85, 145, 365, 256, 31, 20, 33, 50, 87, 149, 375, 625, 512, 48, 21, 34, 51, 89, 173, 425, 725, 1249, 1024, 56, 22, 35, 53, 95, 185, 435, 745, 1489, 3125, 2048, 62, 23, 38, 55, 101, 219, 445, 841, 1825, 3625, 6245

Offset: 0

Antti Karttunen, Jun 28 2020

Array is read by descending antidiagonals with (n,k) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... where A(n,k) is the (k+1)-th solution x to A331410(x) = n. The row indexing (n) starts from 0, and column indexing (k) also from 0.
For any odd prime p that appears on row n, p+1 appears on row n-1.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A331410 is completely additive.

			The top left corner of the array:
  n\k |    0     1     2     3     4     5     6     7     8     9
------+----------------------------------------------------------------
   0  |    1,    2,    4,    8,   16,   32,   64,  128,  256,  512, ...
   1  |    3,    6,    7,   12,   14,   24,   28,   31,   48,   56, ...
   2  |    5,    9,   10,   11,   13,   18,   20,   21,   22,   23, ...
   3  |   15,   17,   19,   27,   30,   33,   34,   35,   38,   39, ...
   4  |   25,   29,   37,   45,   50,   51,   53,   55,   57,   58, ...
   5  |   73,   75,   85,   87,   89,   95,  101,  109,  111,  113, ...
   6  |  125,  145,  149,  173,  185,  219,  225,  250,  255,  261, ...
   7  |  365,  375,  425,  435,  445,  447,  449,  475,  493,  499, ...
   8  |  625,  725,  745,  841,  865,  925,  997, 1009, 1073, 1095, ...
   9  | 1249, 1489, 1825, 1875, 1993, 2017, 2117, 2125, 2175, 2225, ...
etc.

Index entries for sequences that are permutations of the natural numbers

Cf. A331410.
Cf. A329662 (the leftmost column), A000079, A335431, A335882 (rows 0, 1 and 2).
Cf. also A334100 (an analogous array for the map k -> k - k/p), and A335910.

up_to = 78-1; \\ = binomial(12+1,2)-1
memoA331410 = Map();
A331410(n) = if(1==n,0,my(v=0); if(mapisdefined(memoA331410,n,&v), v, my(f=factor(n)); v = sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); mapput(memoA331410,n,v); (v)));
memoA335430sq = Map();
A335430sq(n, k) = { my(v=0); if((0==k), v = -1, if(!mapisdefined(memoA335430sq,[n,k-1],&v), v = A335430sq(n, k-1))); for(i=1+v,oo,if(A331410(1+i)==n,mapput(memoA335430sq,[n,k],i); return(1+i))); };
A335430list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A335430sq(col,(a-(col))))); (v); };
v335430 = A335430list(up_to);
A335430(n) = v335430[1+n];
for(n=0,up_to,print1(A335430(n),", "));

A335881 a(n) = max(A329697(n), A331410(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 2, 1, 2, 2, 3, 0, 3, 2, 3, 2, 3, 2, 3, 1, 4, 2, 3, 2, 4, 3, 3, 0, 3, 3, 3, 2, 4, 3, 3, 2, 3, 3, 4, 2, 4, 3, 4, 1, 4, 4, 4, 2, 4, 3, 4, 2, 4, 4, 4, 3, 3, 3, 4, 0, 4, 3, 4, 3, 4, 3, 4, 2, 5, 4, 5, 3, 4, 3, 4, 2, 4, 3, 3, 3, 5, 4, 5, 2, 5, 4, 4, 3, 4, 4, 5, 1, 3, 4, 4, 4, 5, 4, 3, 2, 4

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

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

Cf. A329697, A331410, A335875, A335877, A335884, A335885, A335904.

Cf. A329662 (apparently the positions of records).

A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A335881(n) = max(A329697(n),A331410(n));

a(n) = max(A329697(n), A331410(n)).

For all n >= 1, A335904(n) >= A335884(n) >= a(n) >= A335875(n) >= A335885(n).

A334099 The least k for which A329697(k) = n; Position of first occurrence of n (and also records) in A329697.

Original entry on oeis.org

1, 3, 7, 19, 43, 127, 283, 659, 1319, 3957, 9227, 21599, 50123, 129263, 258527, 775581, 1551163, 4340087, 9750239, 27353747, 65148847, 156067127, 340997113, 955523423

Offset: 0

Antti Karttunen, Apr 14 2020

Note that although most of the terms after 1 are primes, we also have a few composites: a(9) = a(1)*a(8) = 3*1319 = 3957, a(15) = a(1)*a(14) = 3*258527 = 775581, a(22) = a(8)*a(14) = 340997113.
a(n) <= 3^n and in particular, a(n+1) <= 3*a(n), n > 0 and more generally a(n + m) <= a(n) * a(m) where m, n >= 0. - David A. Corneth, Apr 15 2020
The above follows because A329697 is totally additive.

The leftmost column of A334100.
Cf. A329697 (a left inverse).
Cf. A067513.
Cf. A007755, A105017, and also A329662 (analogous sequence when using the map k -> k + k/p).

With[{s = Array[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 10^6]}, {1}~Join~Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Apr 30 2020 *)

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
m=-1; k=0; for(n=1,2^32, t=A329697(n); if(t>m, m=t; write("b334099.txt", k, " ", n); k++));

For all n >= 0, A329697(a(n)) = n.

cp's OEIS Frontend

A331410 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k + k/p, where p is the largest prime factor of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

Extensions

A335430 Square array where row n lists all numbers k for which A331410(k) = n, read by falling antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A335881 a(n) = max(A329697(n), A331410(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A334099 The least k for which A329697(k) = n; Position of first occurrence of n (and also records) in A329697.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula