A335885 -id:A335885 - cp's OEIS frontend

A335910 Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

1, 2, 3, 4, 5, 9, 8, 6, 11, 27, 16, 7, 13, 33, 81, 32, 10, 15, 37, 99, 243, 64, 12, 18, 39, 107, 297, 729, 128, 14, 19, 43, 109, 321, 891, 2187, 256, 17, 21, 45, 111, 327, 963, 2673, 6561, 512, 20, 22, 53, 117, 333, 981, 2889, 8019, 19683, 1024, 24, 23, 54, 121, 351, 999, 2943, 8667, 24057, 59049, 2048, 28, 25, 55, 129, 363, 1053, 2997, 8829, 26001, 72171, 177147

Offset: 0

Antti Karttunen, Jul 01 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 A335885(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, either p-1 or 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 A335885 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,     5,     6,     7,    10,    12,    14,    17,    20,    24, ...
  2 |     9,    11,    13,    15,    18,    19,    21,    22,    23,    25, ...
  3 |    27,    33,    37,    39,    43,    45,    53,    54,    55,    57, ...
  4 |    81,    99,   107,   109,   111,   117,   121,   129,   131,   135, ...
  5 |   243,   297,   321,   327,   333,   351,   363,   387,   393,   405, ...
  6 |   729,   891,   963,   981,   999,  1053,  1089,  1161,  1177,  1179, ...
  7 |  2187,  2673,  2889,  2943,  2997,  3159,  3267,  3483,  3531,  3537, ...
  8 |  6561,  8019,  8667,  8829,  8991,  9477,  9801, 10449, 10593, 10611, ...
  9 | 19683, 24057, 26001, 26487, 26973, 28431, 29403, 31347, 31779, 31833, ...

Index entries for sequences that are permutations of the natural numbers

Cf. A335885.
Cf. A000079, A335911, A335912 (rows 0-2), A000244 (is very like the leftmost column).
Cf. also arrays A334100, A335430.

up_to = 78-1; \\ = binomial(12+1,2)-1.
memoA335885 = Map();
A335885(n) = if(1==n,0,my(v=0); if(mapisdefined(memoA335885,n,&v), v, my(f=factor(n)); v = sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); mapput(memoA335885,n,v); (v)));
memoA335910sq = Map();
A335910sq(n, k) = { my(v=0); if((0==k), v = -1, if(!mapisdefined(memoA335910sq,[n,k-1],&v), v = A335910sq(n, k-1))); for(i=1+v,oo,if(A335885(1+i)==n,mapput(memoA335910sq,[n,k],i); return(1+i))); };
A335910list(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] = A335910sq(col,(a-(col))))); (v); };
v335910 = A335910list(up_to);
A335910(n) = v335910[1+n];
for(n=0,up_to,print1(A335910(n),", "));

A335912 Numbers k for which A335885(k) = 2.

Original entry on oeis.org

9, 11, 13, 15, 18, 19, 21, 22, 23, 25, 26, 29, 30, 35, 36, 38, 41, 42, 44, 46, 47, 49, 50, 51, 52, 58, 60, 61, 67, 70, 72, 76, 79, 82, 84, 85, 88, 92, 93, 94, 97, 98, 100, 102, 104, 113, 116, 119, 120, 122, 134, 137, 140, 144, 152, 155, 158, 164, 168, 170, 176, 184, 186, 188, 191, 193, 194, 196, 200, 204, 208, 217, 223

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers n such that when we start from k = n, and apply in some combination the nondeterministic maps k -> k - k/p and k -> k + k/p, (where p can be any of the odd prime factors of k), then for some combination we can reach a power of 2 in exactly two steps (but with no combination allowing 0 or 1 steps).

			For n = 70 = 2*5*7, if we first take p = 7 and apply the map n -> n + (n/p), we obtain 80 = 2^4 * 5. We then take p = 5, and apply the map n -> n - (n/p), to obtain 80-16 = 64 = 2^16. Thus we reached a power of 2 in two steps (and there are no shorter paths), therefore 70 is present in this sequence.
For n = 769, which is a prime, 769 - (769/769) yields 768 = 3 * 256. For 768 we can then apply either map to obtain a power of 2, as 768 - (768/3) = 512 = 2^9 and 768 + (768/3) = 1024 = 2^10. On the other hand, 769 + (769/769) = 770 and A335885(770) = 4, so that route would not lead to any shorter paths, therefore 769 is a term of this sequence.

Row 2 of A335910.
Cf. A000265, A335911, A335885.
Subsequences of semiprimes (union gives all odd semiprimes present): A144482, A333788, A336115.
Cf. also A334092, A335874, A336116, A336117 and A334102, A335882, A336122.

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335912(n) = (2==A335885(n));

A335911 Numbers of the form q*(2^k), where k >= 0 and q is either a Fermat prime or a Mersenne prime; Numbers k for which A335885(k) = 1.

Original entry on oeis.org

3, 5, 6, 7, 10, 12, 14, 17, 20, 24, 28, 31, 34, 40, 48, 56, 62, 68, 80, 96, 112, 124, 127, 136, 160, 192, 224, 248, 254, 257, 272, 320, 384, 448, 496, 508, 514, 544, 640, 768, 896, 992, 1016, 1028, 1088, 1280, 1536, 1792, 1984, 2032, 2056, 2176, 2560, 3072, 3584, 3968, 4064, 4112, 4352, 5120, 6144, 7168, 7936, 8128, 8191

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers k such that A000265(k) is either in A000668 or in A019434.

Product of any two terms (whether distinct or not) can be found in A335912.

Row 1 of A335910.
Union of A334101 and A335431. Subsequence of A038550.
Cf. A000668, A000668, A335885, A335912.
Cf. A141453 (after its initial 2, gives the primes present in this sequence).

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335911(n) = (1==A335885(n));

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2, n-1));
isA335911(n) = (isA000668(A000265(n))||isA019434(A000265(n)));

A335907 Average of those twin prime pairs p, q=p+2, for which A335885(p+1) <= A335885(p-1) and A335885(q-1) <= A335885(q+1).

Original entry on oeis.org

4, 12, 72, 102, 108, 180, 192, 240, 348, 420, 600, 822, 828, 1032, 1050, 1152, 1302, 1320, 1428, 1488, 1608, 1722, 1872, 2088, 2112, 2550, 2592, 2688, 2712, 3000, 3168, 3252, 3360, 3372, 3468, 3528, 3672, 3768, 4020, 4092, 4128, 4272, 4650, 4800, 4932, 5100, 5232, 5280, 5520, 5640, 5868, 5880, 6132, 6690, 6762, 6780

Offset: 1

Antti Karttunen, Jul 01 2020

nonn

For such twin prime pairs p, q, A335885(p) = A335885(q) = 1 + A335885((p+q)/2).

Cf. A335885.

Subsequence of A014574. Cf. also A335908.

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335907(n) = (isprime(n-1)&&isprime(n+1)&&(A335885(n)<=A335885(n-2))&&(A335885(n)<=A335885(n+2)));

A329697 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.

Original entry on oeis.org

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

Offset: 1

Ali Sada and Robert G. Wilson v, Feb 28 2020

From Antti Karttunen, Apr 07 2020: (Start)
Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).
Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.
The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.
For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.
(End)
A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is {12, 8}, taking 2 iterations to reach 8 = 2^3. So a(15) is 2.
From _Antti Karttunen_, Apr 07 2020: (Start)
Considering all possible paths from 15 to 1 nondeterministic map k -> k-(k/p), where p can be any prime factor of k, we obtain the following graph:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.
It can be seen that there's also alternative route to 8 via 10 (with 10 = 15-(15/3), where 3 is not the largest prime factor of 15), but it's not any shorter than the route via 12.
(End)

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

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for positions of 0 .. 6 in this sequence, and also array A334100.
Cf. A334099 (a right inverse, positions of the first occurrence of each n).
Cf. A334091 (first differences), A335429 (partial sums).
Cf. also A331410 (analogous sequence when using the map k -> k + k/p), A334861, A335877 (their sums and differences), see also A335878 and A335884, A335885.

a[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] -1; Array[a, 100]

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 07 2020

up_to = 2^24;
A329697list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = if(!bitand(n,n-1),0,1+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v329697 = A329697list(up_to);
A329697(n) = v329697[n]; \\ Antti Karttunen, Apr 07 2020

A329697(n) = if(n<=2,0, if(isprime(n), A329697(n-1)+1, my(f=factor(n)); (apply(A329697, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, Apr 19 2020

From Antti Karttunen, Apr 07-19 2020: (Start)
a(1) = a(2) = 0; and for n > 2, a(p) = 1 + a(p-1) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [This is otherwise equal to the definition of A064097, except here we have a different initial condition, with a(2) = 0].
a(2n) = a(A000265(n)) = a(n).
a(p) = 1+a(p-1), for all odd primes p.
If A209229(n) == 1 [when n is a power of 2], a(n) = 0,
  otherwise a(n) = 1 + a(n-A052126(n)) = 1 + a(A171462(n)).
Equivalently, for non-powers of 2, a(n) = 1 + a(n-(n/A078701(n))),
or equivalently, for non-powers of 2, a(n) = 1 + Min a(n - n/p), for p prime and dividing n.
a(n) = A064097(n) - A064415(n), or equally, a(n) = A064097(n) - A054725(n), for n > 1.
a(A019434(n)) = 1, a(A334092(n)) = 2, a(A334093(n)) = 3, etc. for all applicable n.
For all n >= 0, a(A334099(n)) = a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = a(257^n) = a(65537^n) = n.
a(A122111(n)) = A334107(n), a(A225546(n)) = A334109(n).
(End)
From Antti Karttunen, Mar 16 2021: (Start)
a(n) = a(A336466(n)) + A087436(n) = A336396(n) + A087436(n).
a(A053575(n)) = A336469(n) = a(n) - A005087(n).
a(A147545(n)) = A000120(A147545(n)) - 1.
(End)

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.

Original entry on oeis.org

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

A335884 The length of a longest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

The length of a longest path from n to a power of 2, when using the transitions x -> A171462(x) and x -> A335876(x).

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

Cf. A052126, A171462, A335875, A335876, A335881, A335885, A335904, A335908.

Cf. A335883 (position of the first occurrence of each n).

A335884(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+max(A335884(f[k,1]-1),A335884(f[k,1]+1))))); };

\\ Or empirically as:
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
A209229(n) = (n && !bitand(n,n-1));
A335884(n) = if(A209229(n),0,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, newxs=Set([]); if(!#xs, return(k-1)); for(i=1,#xs,u = xs[i]; a = A171462(u); if(!A209229(a), newxs = setunion([a],newxs)); b = A335876(u); if(!A209229(b), newxs = setunion([b],newxs))); xs = newxs));

Fully additive with a(2) = 0, and a(p) = 1+max(a(p-1), a(p+1)), for odd primes p.
For all n >= 1, A335904(n) >= a(n) >= A335881(n) >= A335875(n) >= A335885(n).
For all n >= 0, a(A335883(n)) = n.

A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 4, 1, 4, 2, 3, 0, 3, 2, 5, 2, 3, 4, 6, 1, 4, 4, 3, 2, 6, 3, 4, 0, 5, 3, 4, 2, 8, 5, 5, 2, 6, 3, 8, 4, 4, 6, 8, 1, 4, 4, 4, 4, 8, 3, 6, 2, 6, 6, 10, 3, 8, 4, 4, 0, 6, 5, 9, 3, 7, 4, 7, 2, 11, 8, 5, 5, 6, 5, 8, 2, 4, 6, 10, 3, 5, 8, 7, 4, 9, 4, 6, 6, 5, 8, 7, 1, 6, 4, 6, 4, 9, 4, 9, 4, 5

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

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

Cf. A000244, A052126, A087436, A171462, A335875, A335876, A335881, A335884, A335885, A335905, A335906, A335915, A336118.

A335904(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A335904(f[k,1]-1)+A335904(f[k,1]+1)))); };

Totally additive with a(2) = 0, and for odd primes p, a(p) = 1 + a(p-1) + a(p+1).
a(n) = A336118(n) +  A087436(n).
For all n >= 1, a(A335915(n)) = A336118(n).
For all n >= 1, a(n) >= A335884(n) >= A335881(n) >= A335875(n) >= A335885(n).
For all n >= 0, a(3^n) = n.

A335875 a(n) = min(A329697(n), A331410(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

			A329697(67) = A331410(67) = 4, thus a(67) = min(4,4) = 4.
A329697(15749) = 9 and A331410(15749) = 10, thus a(15749) = 9.

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

Cf. A329697, A331410, A334861, A335877, A335881, A335884, A335885, A335904.

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)))); };
A335875(n) = min(A329697(n),A331410(n));

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

For all n >= 1, A335904(n) >= A335884(n) >= A335881(n) >= a(n) >= A335885(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).

cp's OEIS Frontend

A335910 Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A335912 Numbers k for which A335885(k) = 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A335911 Numbers of the form q*(2^k), where k >= 0 and q is either a Fermat prime or a Mersenne prime; Numbers k for which A335885(k) = 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

A335907 Average of those twin prime pairs p, q=p+2, for which A335885(p+1) <= A335885(p-1) and A335885(q-1) <= A335885(q+1).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A329697 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

Mathematica

PARI

PARI

PARI

Formula

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

A335884 The length of a longest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

Views