A302774 -id:A302774 - cp's OEIS frontend

A303749 First differences of A302774; Number of terms in A303762 that have prime(n) as their largest prime factor (A006530).

1, 2, 4, 7, 16, 19, 52, 55, 160, 163, 484, 487, 1456, 1459, 4372, 4375, 13120, 13123, 39364, 39367, 118096, 118099, 354292, 354295

Offset: 1

Antti Karttunen, May 05 2018

nonn

For n >= 1, the difference A000079(n-1) - a(n): 0, 0, 0, 1, 0, 13, 12, 73, 96, 349, 540, 1561, 2640, 6733, 12012, 28393, 52416, 117949, 222780, 484921, 930480, 1979053, 3840012, ..., indicates how many squarefree numbers A303762 misses in each round. The first of these is 70 missed at the round 4.

The first differences of these terms is: 1, 2, 3, 9, 3, 33, 3, 105, 3, 321, 3, 969, 3, 2913, 3, 8745, 3, 26241, 3, 78729, 3, 236193, 3, ... which after the first two initial terms seem to be an interleaving of sequences A010701 and A036543.

Cf. A010701, A036543, A302774, A303762.

a(n) = A302774(n+1) - A302774(n).

A303762 a(0) = 1, and for n >= 1, a(n) is either the largest divisor of a(n-1) not already present in the sequence, or (if all divisors already used), a(n-1) * {the least prime p such that p does not divide a(n-1) and p*a(n-1) is not already present}.

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 10, 30, 210, 105, 35, 7, 14, 42, 21, 231, 77, 11, 22, 66, 33, 165, 55, 110, 330, 2310, 1155, 385, 770, 154, 462, 6006, 3003, 1001, 143, 13, 26, 78, 39, 195, 65, 130, 390, 2730, 1365, 455, 91, 182, 546, 273, 4641, 1547, 221, 17, 34, 102, 51, 255, 85, 170, 510, 3570, 1785, 595, 119, 238, 714, 357, 3927, 1309, 187, 374, 1122, 561, 2805, 935

Offset: 0

Antti Karttunen, May 03 2018

nonn

Each a(n+1) is either a divisor or a multiple of a(n).
The construction is otherwise like that of A303760, except here we choose the largest divisor instead of the smallest one. In contrast to A303760, this sequence is NOT permutation of A005117: 70 = A019565(13) is the first missing squarefree number. See also comments in A303769, A303749 and A302775.
Index of greatest prime factor of a(n) is monotonic and increments at n = {0, 1, 2, 4, 8, 15, 31, 50, 102, 157, 317, 480, 964, 1451, 2907, 4366, 8738, 13113, 26233, 39356, ...} - Michael De Vlieger, May 22 2018

			From _Michael De Vlieger_, May 23 2018: (Start)
Table below shows the initial 31 terms at right. First column is index n. Second shows "." if a(n) = largest divisor of a(n-1), or factor p. Third shows presence "1" or absence "." of prime k among prime divisors of a(n).
   n   p\d    MN(n)      a(n)
  ---------------------------
   0     .    .            1
   1     2    1            2
   2     3    11           6
   3     .    .1           3
   4     5    .11         15
   5     .    ..1          5
   6     2    1.1         10
   7     3    111         30
   8     7    1111       210
   9     .    .111       105
  10     .    ..11        35
  11     .    ...1         7
  12     2    1..1        14
  13     3    11.1        42
  14     .    .1.1        21
  15    11    .1.11      231
  16     .    ...11       77
  17     .    ....1       11
  18     2    1...1       22
  19     3    11..1       66
  20     .    .1..1       33
  21     5    .11.1      165
  22     .    ..1.1       55
  23     2    1.1.1      110
  24     3    111.1      330
  25     7    11111     2310
  26     .    .1111     1155
  27     .    ..111      385
  28     2    1.111      770
  29     .    1..11      154
  30     3    11.11      462
  31    13    11.111    6006
  ...  (End)

Antti Karttunen, Table of n, a(n) for n = 0..26232

Subset of A005117.
Cf. also A019565, A302774, A302775, A303749, A303769.
Cf. A303760, A303761 (variants).

Nest[Append[#, Block[{d = Divisors@ #[[-1]], p = 2}, If[Complement[d, #] != {}, Complement[d, #][[-1]], While[Nand[Mod[#[[-1]], p] != 0, FreeQ[#, p #[[-1]] ] ], p = NextPrime@ p]; p #[[-1]] ] ] ] &, {1}, 75] (* Michael De Vlieger, May 22 2018 *)

default(parisizemax,2^31);
up_to = 2^14;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303762 = vector(up_to);
m_inverses = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m_inverses,(prev/d)),v303762[n] = (prev/d);mapput(m_inverses,(prev/d),n);break)); if(!v303762[n], apu = prev; while(mapisdefined(m_inverses,try = prev*A053669(apu)), apu *= A053669(apu)); v303762[n] = try; mapput(m_inverses,try,n)); prev = v303762[n]);
A303762(n) = v303762[n+1];

a(n) = A019565(A303769(n)). [Conjectured]

A303769 a(0) = 0, a(n+1) is either the largest number obtained from a(n) by toggling a single 1-bit off (to 0) if no such number is yet in the sequence, otherwise the least number not yet in sequence that can be obtained from a(n) by toggling a single 0-bit on (to 1). In both cases the bit to be toggled is the rightmost possible that results yet an unencountered number.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 5, 7, 15, 14, 12, 8, 9, 11, 10, 26, 24, 16, 17, 19, 18, 22, 20, 21, 23, 31, 30, 28, 29, 25, 27, 59, 58, 56, 48, 32, 33, 35, 34, 38, 36, 37, 39, 47, 46, 44, 40, 41, 43, 42, 106, 104, 96, 64, 65, 67, 66, 70, 68, 69, 71, 79, 78, 76, 72, 73, 75, 74, 90, 88, 80, 81, 83, 82, 86, 84, 85, 87, 95, 94, 92, 93, 89, 91, 123, 122, 120, 112, 113, 97, 99

Offset: 0

Antti Karttunen, May 03 2018, with more direct definition from David A. Corneth, May 05 2018

The original, but now conjectural, alternative definition is:
a(0) = 0 and for n > 0, if there are one or more k_i that are not already present in the sequence among terms a(0) .. a(n-1), and for which bitor(k_i,a(n-1)) = a(n-1), then a(n) = that k_i which gives maximal value of A019565(k_i) amongst them; otherwise, when no such k_i exists, a(n) = the least number not already present that can be obtained by toggling a single 0-bit of a(n-1) to 1. This is done by trying to toggle successive vacant bits from the least significant end of the binary representation of a(n-1), until such a sum a(n-1) + 2^h (= a(n-1) bitxor 2^h) is found that is not already present in the sequence.
The above construction is otherwise identical to that of A303767, except that we choose k_i with the maximal instead of minimal value of A019565.
In contrast to A303767, this sequence is not surjective (and thus not a permutation of nonnegative integers). The first missing term is 13 = A048675(70). See also comments in A303762, A303749 and A302775.
From David A. Corneth, May 05 2018: (Start)
Another description: a(0) = 0. a(n + 1) is the largest a(n) - 2^j > 0 that's not already in the sequence. If no such value exists, a(n + 1) is the least a(n) + 2^j not already in the sequence.
Using this definition we can prove that 13 isn't in the sequence. (End)
The equivalence of these definitions is still conjectural. The official definition of this sequence follows the latter one. - Antti Karttunen, Jun 08 2018

Antti Karttunen, Table of n, a(n) for n = 0..26232
Index entries for sequences related to binary expansion of n

Cf. A000120, A019565, A048675, A302774, A302775, A303749, A303762, A304749 (terms shown in base-2).

Cf. A303767 (a variant).

Nest[Append[#1, Min@ Select[{#2, #3, 2^IntegerLength[Last@ #1, 2] + Last@ #1}, IntegerQ]] & @@ Function[{a, d}, {a, SelectFirst[Sort@ Map[FromDigits[ReplacePart[d, First@ # -> 1], 2] &, Position[d, 0]], FreeQ[a, #] &], SelectFirst[Sort[#, Greater] &@ Map[FromDigits[ReplacePart[d, First@ # -> 0], 2] &, Position[d, 1]], FreeQ[a, #] &]}] @@ {#, IntegerDigits[Last@ #, 2]} &, {0}, 90] (* Michael De Vlieger, Jun 11 2018 *)

prepare_v303769(up_to) = { my(v = vector(up_to), occurred = Map(), prev=0, b); mapput(occurred,0,0); for(n=1,up_to, b=1; while(b<=prev, if(bitand(prev,b) && !mapisdefined(occurred,prev-b), v[n] = prev-b; break, b <<= 1)); if(!v[n], b=1; while(bitand(prev,b) || mapisdefined(occurred,prev+b), b <<= 1); v[n] = prev+b); mapput(occurred,prev = v[n],n)); (v); };
v303769 = prepare_v303769(16384);
A303769(n) = if(!n,n,v303769[n]); \\ Antti Karttunen, Jun 08 2018

a(n) = A048675(A303762(n)). [The original definition, now conjectured]
For n >= 0, A007088(a(n)) = A304749(n).
From David A. Corneth, May 05 2018: (Start)
The number of ones in the binary expansion of a(n) and a(n + 1) differ by 1. So A000120(a(n)) = A000120(a(n + 1)) +- 1. Furthermore, a(n + 1) <= 3 * a(n).
The number of binary digits of a(n + 1) is 0 or 1 more than the number of binary digits of a(n). So A070939(a(n + 1)) = A070939(a(n)) + 0 or 1. (End)

A302775 a(n) is the first term in A303762 that has prime(n) as one of its prime factors.

Original entry on oeis.org

2, 6, 15, 210, 231, 6006, 4641, 176358, 119301, 6919458, 4663113, 345070362, 228191691, 19624485426, 11248180671, 1192307151126, 748362999111, 91300285891542, 51839992836723, 7361278982814666, 4010248998100527, 633619341699883266, 360208255897878843, 64117069549822434054, 37465998471884193393

Offset: 1

Antti Karttunen, May 05 2018

nonn

It seems that A002110(2n+1)/a(2n+1) = A066205(n).

Cf. A002110, A066205, A302774, A303762.

PARI
```
\\ See program in A302774.
```

a(n) = A303762(A302774(n)).

cp's OEIS Frontend

A303749 First differences of A302774; Number of terms in A303762 that have prime(n) as their largest prime factor (A006530).

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A303762 a(0) = 1, and for n >= 1, a(n) is either the largest divisor of a(n-1) not already present in the sequence, or (if all divisors already used), a(n-1) * {the least prime p such that p does not divide a(n-1) and p*a(n-1) is not already present}.

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A302775 a(n) is the first term in A303762 that has prime(n) as one of its prime factors.

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