A280741 -id:A280741 - cp's OEIS frontend

A372058 Record high-points in A280741.

1, 2, 4, 7, 11, 14, 23, 24, 28, 31, 35, 39, 49, 55, 63, 66, 70, 87, 91, 95, 104, 121, 128, 132, 136, 140, 145, 149, 162, 166, 186, 198, 212, 256, 259, 262, 263, 276, 287, 291, 301, 312, 320, 331, 335, 351, 355, 359, 368, 376, 380, 400, 415, 421, 428, 444, 448, 454, 458

Offset: 1

N. J. A. Sloane, Apr 30 2024

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..5177
N. J. A. Sloane, Graph of A372058 as a function of A372059 [This is the reluctance function for A280864.]

Cf. A280864, A280741, A372059.

A372059 Indices of record high-points in A280741.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

N. J. A. Sloane, Apr 30 2024

nonn

These are the numbers that take the longest to appear in A280864.

It appears to consist of 1, 25, and the primes.

N. J. A. Sloane, Table of n, a(n) for n = 1..5177

Cf. A280864, A280741, A372058.

A372064 a(n) = A280741(n) - A304741(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -3, 0, 0, 0, 1, 0, -3, 0, 5, 0, 5, 1, 0, 0, 5, 0, -5, 9, 9, -3, 0, 5, -32, 5, 5, 0, -1, 5, -2, 6, -14, 5, -23, 5, 3, 9, -18, 5, 13, 5, 12, 1, 5, 5, -26, -23, 13, -1, -2, 5, 12, 5, -1, 5, 4, -4, 9, 6, 5, -16, 13, 5, 17, 7, 6, -2, 5, 9

Offset: 1

N. J. A. Sloane, May 09 2024

sign

A372063 compares A280864 and A280866 term-by-term; the present sequence compares their inverses; and A372065 compares where the primes appear.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A280864, A280741, A280866, A304741, A372063, A372065.

A280864 Lexicographically earliest infinite sequence of distinct positive terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 14, 16, 11, 22, 18, 15, 20, 24, 21, 28, 26, 13, 17, 34, 30, 45, 19, 38, 32, 23, 46, 36, 27, 25, 35, 42, 48, 29, 58, 40, 55, 33, 39, 52, 44, 77, 49, 31, 62, 50, 65, 78, 54, 37, 74, 56, 63, 51, 68, 60, 75, 41, 82, 64, 43, 86

Offset: 1

Rémy Sigrist, Jan 09 2017

In other words, each multiple of a prime p has exactly one neighbor that is also a multiple of p.
This sequence is similar to A280866; the first difference occurs at n=42: a(42)=55 whereas A280866(42)=50.
Conjectured to be a permutation of the positive integers.
Sometimes referred to as the "cup of coffee" sequence, since it feels as if just one more cup of coffee is all it would take to prove that this is indeed a permutation of the positive integers. - N. J. A. Sloane, Nov 04 2020
There are several short cycles, and apparently at least two infinite cycles. For a list see the attached file "Properties of A280864". - N. J. A. Sloane, Feb 03 2017
Properties (For proofs, see the attached file "Properties of A280864")
Theorem 1: This sequence contains every prime and every even number. (Added by N. J. A. Sloane, Jan 15 2017)
Theorem 2: The sequence contains infinitely many odd composite numbers. (Added by N. J. A. Sloane, Feb 14 2017)
Theorem 3: If p is an odd prime, the sequence contains infinitely many odd multiples of p. (Added by N. J. A. Sloane, Mar 12 2017, with corrected proof Apr 03 2017)
There are two types of primes in this sequence: Type I, the first time a term a(n) is divisible by p is when a(n)=p for some n; Type II, the first time a term a(n) is divisible by p is when a(n)=k*p for some n and some k>1 (the Type II primes are listed in A280745).
Conjecture 4: If a prime p divides a(n) then p <= n. - N. J. A. Sloane, Apr 07 2017 and Apr 16 2017
Theorem 5: The sequence is a permutation of the natural numbers iff it contains every square. - N. J. A. Sloane, Apr 14 2017
From Bob Selcoe, Apr 03 2017: (Start)
Define the "radical class" C_R to be the set of numbers which have the same radical R (or the same largest squarefree divisor - i.e., the same product of their prime factors). These are the columns in A284311. So for example C_10 is the set of numbers with radical 10 or prime factors {2,5}: {10, 20, 40, 50, 80, 100, 160, ...}.
If the sequence contains any members of C_R, then those members must appear in order; so for example, if 160 has appeared, {10, 20, 40, 50, 80} will have already appeared, in that order. Naturally, this holds for prime powers; for example, C_5: if 3125 has appeared, {5, 25, 125, 625} will have appeared earlier, in that order.
After seeing a(n), let S be smallest missing number (A280740) and let prime(G) be largest prime already appearing in the sequence. Conjecture: Prime(G) < S <= prime(G+1), and a(35) = 25 = S is the only nonprime S term (following a(31) = 23, preceding a(39) = 29). (End)

			The first terms, alongside their required and forbidden prime factors are:
n   a(n)  Required  Forbidden
--  ----  --------  ---------
1      1  none      none
2      2  none      none
3      4  2         none
4      3  none      2
5      6  3         none
6      8  2         3
7      5  none      2
8     10  5         none
9     12  2         5
10     9  3         2
11     7  none      3
12    14  7         none
13    16  2         7
14    11  none      2
15    22  11        none
16    18  2         11
17    15  3         2
18    20  5         3
19    24  2         5
20    21  3         2
21    28  7         3
22    26  2         7
23    13  13        2
24    17  none      13
25    34  17        none
26    30  2         17
27    45  3, 5      2
28    19  none      3, 5
29    38  19        none
30    32  2         19
31    23  none      2
32    46  23        none
33    36  2         23
34    27  3         2
35    25  none      3
36    35  5         none
37    42  7         5
38    48  2, 3      7
39    29  none      2, 3
40    58  29        none
41    40  2         29
42    55  5         2

N. J. A. Sloane, Table of n, a(n) for n = 1..100000 (First 10000 terms from Rémy Sigrist)
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
Rémy Sigrist, PARI program for A280864
N. J. A. Sloane, Properties of A280864 [Revised, Apr 25 2017]
N. J. A. Sloane, Table of n, a(n) for n = 1..1000000, computed using Sigrist's PARI program.
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.

See A280738, A280740, A280741 (inverse), A280742, A280743, A280744, A280745, A280746, A280755, A280770, A280771, A280773, A280774, A283832, A284724, A284725, A284726, A284785, A285181 for various subsidiary sequences.
A280754 gives fixed points.
Cf. A280866.
In the same spirit as A064413 and A098550.
A338338, A338444, and A375029 are variants.
A373797 is a finite version.

N:= 1000: # to get all terms until the first term > N
A[1]:= 1:
A[2]:= 2:
G:= {}:
Avail:= [$3..N]:
found:= true:
lastn:= 2:
for n from 3 while found and nops(Avail)>0 do
  found:= false;
  H:= G;
  G:= numtheory:-factorset(A[n-1]);
  r:= convert(G minus H,`*`);
  s:= convert(G intersect H, `*`);
  for j from 1 to nops(Avail) do
    if Avail[j] mod r = 0 and igcd(Avail[j],s) = 1 then
      found:= true;
      A[n]:= Avail[j];
      Avail:= subsop(j=NULL,Avail);
      lastn:= n;
      break
    fi
  od;
od:
seq(A[i],i=1..lastn); # Robert Israel, Mar 22 2017

terms = 100;
rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]] (* Jean-François Alcover, Nov 23 2017, translated from Rémy Sigrist's PARI program *)

Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019

A280742 Position where n-th prime appears in A280864.

Original entry on oeis.org

2, 4, 7, 11, 14, 23, 24, 28, 31, 39, 49, 55, 63, 66, 70, 87, 91, 95, 104, 121, 128, 132, 136, 140, 145, 149, 162, 166, 186, 198, 212, 256, 259, 262, 263, 276, 287, 291, 301, 312, 320, 331, 335, 351, 355, 359, 368, 376, 380, 400, 415, 421, 428, 444, 448, 454

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

This sequence is very similar to A372058, the difference being caused by the presence of 1 and 25 in A372059. - N. J. A. Sloane, Jun 06 2024

David Suteu and N. J. A. Sloane, Table of n, a(n) for n = 1..5175 [First 679 terms from David Suteu]

Cf. A280864, A280741, A280745, A372058, A372059..

a(9) inserted and terms a(11) and beyond added by Daniel Suteu, Jan 13 2017

A280738 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives p(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 1, 11, 2, 3, 5, 2, 3, 7, 2, 13, 1, 17, 2, 15, 1, 19, 2, 1, 23, 2, 3, 1, 5, 7, 6, 1, 29, 2, 5, 11, 3, 13, 2, 11, 7, 1, 31, 2, 5, 13, 6, 1, 37, 2, 7, 3, 17, 2, 15, 1, 41, 2, 1, 43, 2, 33, 1, 47, 2, 35, 3, 19, 2, 3, 23, 2, 5

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

We use the convention that an empty product is 1.
By decree, gcd(S(n+1),p(n)) = 1, gcd(S(n+1),q(n)) = q(n) = p(n+1), S(n+1) >= r(n).
By definition, all terms are squarefree. Let {i,j,k} be distinct fixed positive numbers. Conjecture: All squarefree numbers appear infinitely often, and all terms a(n) = j are immediately preceded and followed infinitely often by all terms a(n-1) = i and a(n+1) = k. If so, then A280864 is a permutation of the natural numbers. - Bob Selcoe, Apr 04 2017

Rémy Sigrist, Table of n, a(n) for n = 1..100000

Cf. A280864, A280740, A280741, A280742, A280743, A280744.

Cf. A005117 (squarefree numbers).

More terms from Rémy Sigrist, Jan 14 2017

A280740 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives r(n).

Original entry on oeis.org

2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 19, 19, 19, 19, 23, 23, 23, 25, 25, 25, 25, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 41, 41, 43, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

We use the convention that an empty product is 1.
By decree, gcd(S(n+1),p(n)) = 1, gcd(S(n+1),q(n)) = q(n) = p(n+1), S(n+1) >= r(n). (Note p(n) is as defined above; it is not the n-th prime.)
Conjecture: except for the four terms equal to 25, a(n) is always a prime, and all the primes appear and in their natural order.
The conjecture is true for n up to 10^7. - Lars Blomberg Jan 14 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A280864, A280738, A280741, A280742, A280743, A280744.

A280743 Records in A280864.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 22, 24, 28, 34, 45, 46, 48, 58, 77, 78, 82, 86, 99, 105, 121, 135, 136, 165, 195, 325, 455, 459, 465, 605, 637, 651, 715, 777, 897, 987, 1001, 1495, 1573, 2275, 2387, 2415, 2421, 2451, 2493, 2919, 3129, 3171, 3633, 3689, 4011, 4053, 4137, 4179, 4879, 4893

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1289 [First 226 terms from N. J. A. Sloane]

Cf. A280864, A280738, A280740, A280741, A280742, A280744.

A280744 Positions of records in A280864.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 13, 15, 19, 21, 25, 27, 32, 38, 40, 47, 53, 64, 67, 69, 73, 86, 90, 99, 107, 114, 131, 189, 290, 304, 311, 354, 366, 387, 435, 520, 553, 563, 598, 817, 885, 1333, 1361, 1615, 1634, 1651, 1655, 1765, 1776, 2031, 2068, 2248, 2261, 2314, 2700, 2704, 2712, 2993, 3128, 3670

Offset: 1

N. J. A. Sloane, Jan 12 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1289 (First 226 terms from N. J. A. Sloane)

Cf. A280864, A280738, A280740, A280741, A280742, A280743.

A304741 Inverse permutation to A280866.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 11, 6, 10, 8, 14, 9, 23, 12, 17, 13, 24, 16, 28, 18, 20, 15, 31, 19, 35, 22, 34, 21, 39, 26, 43, 30, 46, 25, 36, 33, 54, 29, 47, 41, 58, 37, 61, 45, 27, 32, 65, 38, 53, 42, 50, 48, 87, 49, 74, 52, 69, 40, 92, 56, 97, 44, 72, 60, 75, 63, 101, 51, 95, 67, 108, 71, 116, 55, 57, 70, 73, 76, 119

Offset: 1

Antti Karttunen, May 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences that are permutations of the natural numbers

Cf. A280866 (inverse).

Differs from similar A280741 for the first time at n=31, where a(31) = 43, while A280741(31) = 49.

PARI
```
\\ Use the program given in A280866.
```

cp's OEIS Frontend

A372058 Record high-points in A280741.

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A372059 Indices of record high-points in A280741.

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A372064 a(n) = A280741(n) - A304741(n).

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A280864 Lexicographically earliest infinite sequence of distinct positive terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2.

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A280742 Position where n-th prime appears in A280864.

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A280738 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives p(n).

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A280740 After S(n)=A280864(n) has been computed, let p(n) = product of distinct primes shared by S(n-1) and S(n); let q(n) = product of distinct primes in S(n) but not in S(n-1); and let r(n) = smallest number not yet in S. Sequence gives r(n).

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A280743 Records in A280864.

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A280744 Positions of records in A280864.

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A304741 Inverse permutation to A280866.

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PARI